9 11

9_10

9_12

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 11's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 11 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8
Gauss code	-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 10 14 16 12 2 18 6 8
Conway Notation	[4122]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{11, 7}, {8, 6}, {7, 10}, {1, 8}, {9, 11}, {10, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 1}, {2, 9}]

[edit Notes on presentations of 9 11]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 11"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 12 2 18 6 8

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[4122]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(4,\{1,1,1,1,-2,1,3,-2,3\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 4, 9, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{11, 7}, {8, 6}, {7, 10}, {1, 8}, {9, 11}, {10, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 1}, {2, 9}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [1][-12] Hyperbolic Volume 8.28859 A-Polynomial See Data:9 11/A-polynomial

[edit Notes for 9 11's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant 4

[edit Notes for 9 11's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-7t+7-7t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, 4 }
Jones polynomial	$-q^{9}+2q^{8}-4q^{7}+5q^{6}-5q^{5}+6q^{4}-4q^{3}+3q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-4z^{4}a^{-4}+2z^{4}a^{-6}+3z^{2}a^{-2}-4z^{2}a^{-4}+6z^{2}a^{-6}-z^{2}a^{-8}+a^{-2}-a^{-4}+3a^{-6}-2a^{-8}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+4z^{7}a^{-5}+2z^{7}a^{-7}+z^{6}a^{-2}-z^{6}a^{-4}+z^{6}a^{-6}+3z^{6}a^{-8}-8z^{5}a^{-3}-12z^{5}a^{-5}-z^{5}a^{-7}+3z^{5}a^{-9}-4z^{4}a^{-2}-5z^{4}a^{-4}-7z^{4}a^{-6}-4z^{4}a^{-8}+2z^{4}a^{-10}+8z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}+4z^{2}a^{-2}+5z^{2}a^{-4}+6z^{2}a^{-6}+4z^{2}a^{-8}-z^{2}a^{-10}-za^{-3}-2za^{-5}+2za^{-7}+2za^{-9}-za^{-11}-a^{-2}-a^{-4}-3a^{-6}-2a^{-8}$
The A2 invariant	$1-q^{-8}+2q^{-10}+2q^{-14}+q^{-16}+q^{-20}-q^{-22}-q^{-26}-q^{-28}$
The G2 invariant	$q^{-2}-q^{-4}+3q^{-6}-4q^{-8}+3q^{-10}-q^{-12}-2q^{-14}+10q^{-16}-12q^{-18}+13q^{-20}-7q^{-22}-2q^{-24}+10q^{-26}-17q^{-28}+17q^{-30}-12q^{-32}+2q^{-34}+8q^{-36}-13q^{-38}+12q^{-40}-7q^{-42}-2q^{-44}+7q^{-46}-8q^{-48}+4q^{-50}-q^{-52}-5q^{-54}+16q^{-56}-12q^{-58}+10q^{-60}-q^{-62}-8q^{-64}+19q^{-66}-20q^{-68}+18q^{-70}-9q^{-72}+16q^{-76}-19q^{-78}+17q^{-80}-8q^{-82}-2q^{-84}+8q^{-86}-10q^{-88}+4q^{-90}-4q^{-94}+7q^{-96}-6q^{-98}+3q^{-102}-10q^{-104}+10q^{-106}-9q^{-108}+4q^{-110}-q^{-112}-4q^{-114}+8q^{-116}-11q^{-118}+11q^{-120}-7q^{-122}+2q^{-124}+q^{-126}-6q^{-128}+6q^{-130}-6q^{-132}+5q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

Further Quantum Invariants

Further quantum knot invariants for 9_11.

A1 Invariants.

Weight	Invariant
1	$q-q^{-1}+q^{-3}-q^{-5}+2q^{-7}+q^{-9}+q^{-13}-2q^{-15}+q^{-17}-q^{-19}$
2	$q^{6}-q^{4}-2q^{2}+3+q^{-2}-4q^{-4}+3q^{-6}+4q^{-8}-4q^{-10}+q^{-12}+4q^{-14}-4q^{-16}-q^{-18}+4q^{-20}-2q^{-24}+2q^{-26}+3q^{-28}-2q^{-30}-4q^{-32}+4q^{-34}-q^{-36}-5q^{-38}+4q^{-40}-q^{-42}-3q^{-44}+3q^{-46}-q^{-50}+q^{-52}$
3	$q^{15}-q^{13}-2q^{11}+4q^{7}+3q^{5}-5q^{3}-6q+4q^{-1}+9q^{-3}-11q^{-7}-4q^{-9}+11q^{-11}+8q^{-13}-7q^{-15}-10q^{-17}+4q^{-19}+13q^{-21}+q^{-23}-9q^{-25}-3q^{-27}+10q^{-29}+4q^{-31}-8q^{-33}-8q^{-35}+8q^{-37}+8q^{-39}-7q^{-41}-9q^{-43}+4q^{-45}+10q^{-47}-8q^{-51}-6q^{-53}+6q^{-55}+8q^{-57}-q^{-59}-14q^{-61}-2q^{-63}+10q^{-65}+5q^{-67}-9q^{-69}-4q^{-71}+6q^{-73}+2q^{-75}-3q^{-77}+3q^{-81}-q^{-83}-2q^{-85}+2q^{-87}+2q^{-89}-q^{-91}-q^{-93}+q^{-97}-q^{-99}$
4	$q^{28}-q^{26}-2q^{24}+q^{20}+6q^{18}+q^{16}-5q^{14}-7q^{12}-7q^{10}+12q^{8}+13q^{6}+3q^{4}-10q^{2}-24+16q^{-4}+23q^{-6}+9q^{-8}-29q^{-10}-21q^{-12}-4q^{-14}+25q^{-16}+33q^{-18}-5q^{-20}-21q^{-22}-31q^{-24}+35q^{-28}+21q^{-30}+3q^{-32}-36q^{-34}-28q^{-36}+14q^{-38}+30q^{-40}+30q^{-42}-20q^{-44}-37q^{-46}-6q^{-48}+23q^{-50}+36q^{-52}-9q^{-54}-36q^{-56}-13q^{-58}+20q^{-60}+30q^{-62}-7q^{-64}-32q^{-66}-16q^{-68}+19q^{-70}+29q^{-72}+3q^{-74}-25q^{-76}-27q^{-78}+6q^{-80}+22q^{-82}+24q^{-84}+q^{-86}-29q^{-88}-25q^{-90}-6q^{-92}+38q^{-94}+35q^{-96}-11q^{-98}-37q^{-100}-38q^{-102}+22q^{-104}+47q^{-106}+14q^{-108}-18q^{-110}-45q^{-112}-q^{-114}+28q^{-116}+17q^{-118}+5q^{-120}-26q^{-122}-7q^{-124}+7q^{-126}+5q^{-128}+11q^{-130}-8q^{-132}-2q^{-134}-2q^{-136}-3q^{-138}+7q^{-140}-2q^{-142}+q^{-144}-q^{-146}-3q^{-148}+2q^{-150}-q^{-152}+q^{-154}-q^{-158}+q^{-160}$
5	$q^{45}-q^{43}-2q^{41}+q^{37}+3q^{35}+4q^{33}+q^{31}-7q^{29}-9q^{27}-5q^{25}+3q^{23}+15q^{21}+18q^{19}+5q^{17}-18q^{15}-28q^{13}-20q^{11}+5q^{9}+35q^{7}+41q^{5}+16q^{3}-27q-53q^{-1}-43q^{-3}-q^{-5}+51q^{-7}+67q^{-9}+35q^{-11}-27q^{-13}-69q^{-15}-66q^{-17}-16q^{-19}+51q^{-21}+84q^{-23}+59q^{-25}-9q^{-27}-73q^{-29}-89q^{-31}-43q^{-33}+41q^{-35}+100q^{-37}+92q^{-39}+10q^{-41}-83q^{-43}-118q^{-45}-67q^{-47}+48q^{-49}+132q^{-51}+111q^{-53}-7q^{-55}-120q^{-57}-139q^{-59}-44q^{-61}+98q^{-63}+156q^{-65}+73q^{-67}-67q^{-69}-145q^{-71}-97q^{-73}+40q^{-75}+135q^{-77}+98q^{-79}-23q^{-81}-112q^{-83}-94q^{-85}+12q^{-87}+99q^{-89}+81q^{-91}-16q^{-93}-90q^{-95}-73q^{-97}+16q^{-99}+86q^{-101}+69q^{-103}-18q^{-105}-88q^{-107}-75q^{-109}+4q^{-111}+75q^{-113}+87q^{-115}+31q^{-117}-52q^{-119}-90q^{-121}-74q^{-123}+5q^{-125}+91q^{-127}+114q^{-129}+55q^{-131}-56q^{-133}-148q^{-135}-122q^{-137}+18q^{-139}+149q^{-141}+166q^{-143}+51q^{-145}-127q^{-147}-199q^{-149}-96q^{-151}+86q^{-153}+190q^{-155}+132q^{-157}-37q^{-159}-167q^{-161}-141q^{-163}-q^{-165}+127q^{-167}+128q^{-169}+23q^{-171}-82q^{-173}-104q^{-175}-34q^{-177}+51q^{-179}+74q^{-181}+33q^{-183}-25q^{-185}-47q^{-187}-29q^{-189}+8q^{-191}+28q^{-193}+22q^{-195}-q^{-197}-15q^{-199}-14q^{-201}-4q^{-203}+6q^{-205}+9q^{-207}+4q^{-209}-4q^{-211}-3q^{-213}-q^{-215}-q^{-217}+2q^{-219}+2q^{-221}-q^{-223}+q^{-227}-q^{-229}+q^{-233}-q^{-235}$

A2 Invariants.

Weight	Invariant
1,0	$1-q^{-8}+2q^{-10}+2q^{-14}+q^{-16}+q^{-20}-q^{-22}-q^{-26}-q^{-28}$
1,1	$q^{4}-2q^{2}+6-12q^{-2}+19q^{-4}-30q^{-6}+40q^{-8}-44q^{-10}+49q^{-12}-44q^{-14}+38q^{-16}-18q^{-18}+3q^{-20}+18q^{-22}-34q^{-24}+54q^{-26}-68q^{-28}+74q^{-30}-76q^{-32}+74q^{-34}-63q^{-36}+50q^{-38}-32q^{-40}+14q^{-42}-2q^{-44}-10q^{-46}+14q^{-48}-22q^{-50}+25q^{-52}-26q^{-54}+24q^{-56}-28q^{-58}+26q^{-60}-22q^{-62}+18q^{-64}-14q^{-66}+11q^{-68}-6q^{-70}+4q^{-72}-2q^{-74}+q^{-76}$
2,0	$q^{4}-1-q^{-2}+q^{-4}+q^{-6}-2q^{-8}+4q^{-12}+3q^{-14}-q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-24}+q^{-28}-2q^{-30}+3q^{-32}+3q^{-34}+q^{-38}+4q^{-40}-4q^{-44}-q^{-46}-2q^{-50}-4q^{-52}-q^{-54}-q^{-58}+q^{-66}+q^{-68}+q^{-70}$

A3 Invariants.

Weight	Invariant
0,1,0	$1-q^{-2}+q^{-4}+q^{-6}-2q^{-8}+3q^{-10}+q^{-12}-2q^{-14}+3q^{-16}-4q^{-20}+3q^{-22}+3q^{-24}-2q^{-26}+5q^{-28}+4q^{-30}+2q^{-32}-q^{-34}-6q^{-40}-3q^{-42}+2q^{-44}-4q^{-46}-2q^{-48}+5q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-q^{-60}+q^{-62}$
1,0,0	$q^{-1}+q^{-5}-q^{-7}+q^{-9}-q^{-11}+q^{-13}+q^{-17}+q^{-19}+q^{-21}+2q^{-23}+2q^{-27}-q^{-29}-2q^{-33}-q^{-35}-q^{-37}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-8}+q^{-14}+q^{-16}+2q^{-20}+2q^{-22}-q^{-24}-q^{-26}+2q^{-28}+2q^{-30}-4q^{-32}+2q^{-34}+6q^{-36}+2q^{-38}+q^{-40}+7q^{-42}+4q^{-44}+q^{-48}-5q^{-52}-6q^{-54}-2q^{-56}-4q^{-58}-6q^{-60}-q^{-62}+2q^{-64}-2q^{-66}-q^{-68}+2q^{-70}+2q^{-72}+q^{-78}+q^{-80}$
1,0,0,0	$q^{-2}+q^{-6}+q^{-12}-q^{-14}+q^{-16}-q^{-18}+q^{-20}+q^{-24}+q^{-26}+2q^{-28}+2q^{-30}+q^{-32}+2q^{-34}-q^{-36}-2q^{-40}-2q^{-42}-q^{-44}-q^{-46}$

B2 Invariants.

Weight	Invariant
0,1	$1-q^{-2}+3q^{-4}-3q^{-6}+4q^{-8}-5q^{-10}+5q^{-12}-4q^{-14}+3q^{-16}-2q^{-18}+3q^{-22}-5q^{-24}+8q^{-26}-7q^{-28}+10q^{-30}-8q^{-32}+9q^{-34}-6q^{-36}+4q^{-38}-2q^{-40}-q^{-42}+2q^{-44}-4q^{-46}+4q^{-48}-5q^{-50}+4q^{-52}-4q^{-54}+3q^{-56}-2q^{-58}+q^{-60}-q^{-62}$
1,0	$q^{2}-q^{-2}-q^{-4}+2q^{-6}+2q^{-8}-2q^{-10}-3q^{-12}+q^{-14}+5q^{-16}+2q^{-18}-4q^{-20}-3q^{-22}+3q^{-24}+5q^{-26}-5q^{-30}-3q^{-32}+3q^{-34}+4q^{-36}-3q^{-40}+2q^{-42}+4q^{-44}+2q^{-46}-2q^{-48}+q^{-50}+4q^{-52}+2q^{-54}-4q^{-56}-3q^{-58}+2q^{-60}+2q^{-62}-3q^{-64}-5q^{-66}-q^{-68}+3q^{-70}+q^{-72}-4q^{-74}-4q^{-76}+q^{-78}+5q^{-80}+q^{-82}-3q^{-84}-3q^{-86}+q^{-88}+3q^{-90}+q^{-92}-q^{-94}-q^{-96}+q^{-100}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-2}-q^{-4}+2q^{-6}-2q^{-8}+4q^{-10}-3q^{-12}+4q^{-14}-3q^{-16}+5q^{-18}-3q^{-20}+2q^{-22}-2q^{-24}+q^{-26}-2q^{-30}+3q^{-32}-3q^{-34}+7q^{-36}-4q^{-38}+10q^{-40}-3q^{-42}+10q^{-44}-5q^{-46}+7q^{-48}-5q^{-50}+2q^{-52}-5q^{-54}-3q^{-56}-3q^{-58}-3q^{-60}+q^{-62}-4q^{-64}+2q^{-66}-3q^{-68}+5q^{-70}-3q^{-72}+2q^{-74}-3q^{-76}+3q^{-78}-q^{-80}+q^{-82}-q^{-84}+q^{-86}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-2}-q^{-4}+3q^{-6}-4q^{-8}+3q^{-10}-q^{-12}-2q^{-14}+10q^{-16}-12q^{-18}+13q^{-20}-7q^{-22}-2q^{-24}+10q^{-26}-17q^{-28}+17q^{-30}-12q^{-32}+2q^{-34}+8q^{-36}-13q^{-38}+12q^{-40}-7q^{-42}-2q^{-44}+7q^{-46}-8q^{-48}+4q^{-50}-q^{-52}-5q^{-54}+16q^{-56}-12q^{-58}+10q^{-60}-q^{-62}-8q^{-64}+19q^{-66}-20q^{-68}+18q^{-70}-9q^{-72}+16q^{-76}-19q^{-78}+17q^{-80}-8q^{-82}-2q^{-84}+8q^{-86}-10q^{-88}+4q^{-90}-4q^{-94}+7q^{-96}-6q^{-98}+3q^{-102}-10q^{-104}+10q^{-106}-9q^{-108}+4q^{-110}-q^{-112}-4q^{-114}+8q^{-116}-11q^{-118}+11q^{-120}-7q^{-122}+2q^{-124}+q^{-126}-6q^{-128}+6q^{-130}-6q^{-132}+5q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 11"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{3}+5t^{2}-7t+7-7t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-z^{4}+4z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 33, 4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{9}+2q^{8}-4q^{7}+5q^{6}-5q^{5}+6q^{4}-4q^{3}+3q^{2}-2q+1$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{6}a^{-4}+z^{4}a^{-2}-4z^{4}a^{-4}+2z^{4}a^{-6}+3z^{2}a^{-2}-4z^{2}a^{-4}+6z^{2}a^{-6}-z^{2}a^{-8}+a^{-2}-a^{-4}+3a^{-6}-2a^{-8}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+4z^{7}a^{-5}+2z^{7}a^{-7}+z^{6}a^{-2}-z^{6}a^{-4}+z^{6}a^{-6}+3z^{6}a^{-8}-8z^{5}a^{-3}-12z^{5}a^{-5}-z^{5}a^{-7}+3z^{5}a^{-9}-4z^{4}a^{-2}-5z^{4}a^{-4}-7z^{4}a^{-6}-4z^{4}a^{-8}+2z^{4}a^{-10}+8z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}+4z^{2}a^{-2}+5z^{2}a^{-4}+6z^{2}a^{-6}+4z^{2}a^{-8}-z^{2}a^{-10}-za^{-3}-2za^{-5}+2za^{-7}+2za^{-9}-za^{-11}-a^{-2}-a^{-4}-3a^{-6}-2a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n95,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 11"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-t^{3}+5t^{2}-7t+7-7t^{-1}+5t^{-2}-t^{-3}$, $-q^{9}+2q^{8}-4q^{7}+5q^{6}-5q^{5}+6q^{4}-4q^{3}+3q^{2}-2q+1$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n95,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(4, 9)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $16$ $72$ $128$ ${\frac {1160}{3}}$ ${\frac {208}{3}}$ $1152$ $2288$ $384$ $392$ ${\frac {2048}{3}}$ $2592$ ${\frac {18560}{3}}$ ${\frac {3328}{3}}$ ${\frac {204662}{15}}$ $-{\frac {5408}{15}}$ ${\frac {284288}{45}}$ ${\frac {1450}{9}}$ ${\frac {13382}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$4 is the signature of 9 11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 19                   1 -1 17                 1   1 15               3 1   -2 13             2 1     1 11           3 3       0 9         3 2         1 7       1 3           2 5     2 3             -1 3   1 2               1 1   1                 -1 -1 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{25}-2q^{24}+q^{23}+4q^{22}-8q^{21}+3q^{20}+9q^{19}-17q^{18}+7q^{17}+14q^{16}-25q^{15}+9q^{14}+19q^{13}-26q^{12}+5q^{11}+21q^{10}-22q^{9}+18q^{7}-14q^{6}-3q^{5}+13q^{4}-6q^{3}-4q^{2}+6q-1-2q^{-1}+q^{-2}$
3	$-q^{48}+2q^{47}-q^{46}-q^{45}-q^{44}+5q^{43}-q^{42}-5q^{41}+9q^{39}-4q^{38}-8q^{37}+5q^{36}+13q^{35}-14q^{34}-13q^{33}+19q^{32}+18q^{31}-26q^{30}-25q^{29}+32q^{28}+27q^{27}-28q^{26}-37q^{25}+30q^{24}+35q^{23}-18q^{22}-43q^{21}+17q^{20}+37q^{19}-3q^{18}-43q^{17}+q^{16}+37q^{15}+9q^{14}-37q^{13}-12q^{12}+31q^{11}+19q^{10}-25q^{9}-21q^{8}+17q^{7}+22q^{6}-10q^{5}-18q^{4}+2q^{3}+15q^{2}+q-9-3q^{-1}+5q^{-2}+2q^{-3}-q^{-4}-2q^{-5}+q^{-6}$
4	$q^{78}-2q^{77}+q^{76}+q^{75}-2q^{74}+4q^{73}-7q^{72}+3q^{71}+3q^{70}-5q^{69}+13q^{68}-17q^{67}+4q^{66}+3q^{65}-11q^{64}+32q^{63}-23q^{62}+6q^{61}-11q^{60}-30q^{59}+63q^{58}-11q^{57}+17q^{56}-40q^{55}-74q^{54}+90q^{53}+21q^{52}+50q^{51}-65q^{50}-134q^{49}+91q^{48}+47q^{47}+96q^{46}-62q^{45}-178q^{44}+72q^{43}+43q^{42}+126q^{41}-39q^{40}-180q^{39}+56q^{38}+10q^{37}+128q^{36}-11q^{35}-154q^{34}+46q^{33}-25q^{32}+112q^{31}+14q^{30}-117q^{29}+36q^{28}-58q^{27}+89q^{26}+41q^{25}-72q^{24}+23q^{23}-87q^{22}+58q^{21}+58q^{20}-22q^{19}+23q^{18}-103q^{17}+16q^{16}+50q^{15}+17q^{14}+41q^{13}-89q^{12}-19q^{11}+19q^{10}+27q^{9}+57q^{8}-51q^{7}-27q^{6}-10q^{5}+10q^{4}+49q^{3}-13q^{2}-13q-17-6q^{-1}+25q^{-2}+q^{-3}-7q^{-5}-7q^{-6}+6q^{-7}+q^{-8}+2q^{-9}-q^{-10}-2q^{-11}+q^{-12}$
5	$-q^{115}+2q^{114}-q^{113}-q^{112}+2q^{111}-q^{110}-2q^{109}+5q^{108}-q^{107}-4q^{106}+2q^{105}-3q^{104}-3q^{103}+13q^{102}+4q^{101}-7q^{100}-8q^{99}-13q^{98}-4q^{97}+27q^{96}+27q^{95}-q^{94}-28q^{93}-50q^{92}-22q^{91}+49q^{90}+85q^{89}+40q^{88}-51q^{87}-135q^{86}-92q^{85}+71q^{84}+190q^{83}+145q^{82}-52q^{81}-263q^{80}-232q^{79}+45q^{78}+320q^{77}+314q^{76}+6q^{75}-367q^{74}-414q^{73}-58q^{72}+392q^{71}+492q^{70}+121q^{69}-384q^{68}-545q^{67}-198q^{66}+366q^{65}+584q^{64}+232q^{63}-325q^{62}-568q^{61}-284q^{60}+287q^{59}+568q^{58}+270q^{57}-242q^{56}-512q^{55}-296q^{54}+216q^{53}+489q^{52}+257q^{51}-172q^{50}-425q^{49}-279q^{48}+146q^{47}+400q^{46}+240q^{45}-98q^{44}-328q^{43}-261q^{42}+59q^{41}+294q^{40}+222q^{39}-9q^{38}-207q^{37}-224q^{36}-36q^{35}+157q^{34}+174q^{33}+69q^{32}-67q^{31}-141q^{30}-94q^{29}+15q^{28}+79q^{27}+88q^{26}+46q^{25}-23q^{24}-73q^{23}-69q^{22}-36q^{21}+37q^{20}+81q^{19}+70q^{18}+9q^{17}-61q^{16}-95q^{15}-47q^{14}+35q^{13}+86q^{12}+73q^{11}+7q^{10}-70q^{9}-80q^{8}-32q^{7}+36q^{6}+70q^{5}+49q^{4}-8q^{3}-48q^{2}-48q-16+28q^{-1}+39q^{-2}+18q^{-3}-5q^{-4}-23q^{-5}-22q^{-6}-2q^{-7}+14q^{-8}+10q^{-9}+5q^{-10}-9q^{-12}-5q^{-13}+2q^{-14}+2q^{-15}+q^{-16}+2q^{-17}-q^{-18}-2q^{-19}+q^{-20}$
6	$q^{159}-2q^{158}+q^{157}+q^{156}-2q^{155}+q^{154}-q^{153}+4q^{152}-7q^{151}+2q^{150}+7q^{149}-6q^{148}+4q^{147}-3q^{146}+4q^{145}-20q^{144}+5q^{143}+24q^{142}-7q^{141}+12q^{140}-8q^{139}-5q^{138}-54q^{137}+7q^{136}+58q^{135}+7q^{134}+40q^{133}-14q^{132}-37q^{131}-130q^{130}-6q^{129}+116q^{128}+63q^{127}+116q^{126}-12q^{125}-120q^{124}-291q^{123}-64q^{122}+209q^{121}+215q^{120}+297q^{119}+20q^{118}-292q^{117}-608q^{116}-232q^{115}+323q^{114}+511q^{113}+664q^{112}+152q^{111}-524q^{110}-1111q^{109}-601q^{108}+354q^{107}+881q^{106}+1227q^{105}+488q^{104}-661q^{103}-1664q^{102}-1161q^{101}+161q^{100}+1111q^{99}+1803q^{98}+1003q^{97}-535q^{96}-2011q^{95}-1700q^{94}-220q^{93}+1046q^{92}+2132q^{91}+1460q^{90}-213q^{89}-2027q^{88}-1968q^{87}-553q^{86}+788q^{85}+2129q^{84}+1654q^{83}+73q^{82}-1842q^{81}-1932q^{80}-684q^{79}+546q^{78}+1940q^{77}+1608q^{76}+225q^{75}-1630q^{74}-1755q^{73}-687q^{72}+371q^{71}+1717q^{70}+1490q^{69}+337q^{68}-1415q^{67}-1568q^{66}-707q^{65}+179q^{64}+1469q^{63}+1388q^{62}+513q^{61}-1124q^{60}-1363q^{59}-775q^{58}-91q^{57}+1136q^{56}+1257q^{55}+729q^{54}-726q^{53}-1067q^{52}-805q^{51}-391q^{50}+697q^{49}+1007q^{48}+865q^{47}-291q^{46}-644q^{45}-685q^{44}-594q^{43}+226q^{42}+607q^{41}+804q^{40}+36q^{39}-180q^{38}-384q^{37}-572q^{36}-118q^{35}+159q^{34}+522q^{33}+117q^{32}+149q^{31}-21q^{30}-318q^{29}-196q^{28}-138q^{27}+171q^{26}-44q^{25}+198q^{24}+193q^{23}-8q^{22}-42q^{21}-155q^{20}-24q^{19}-240q^{18}+32q^{17}+150q^{16}+133q^{15}+124q^{14}+5q^{13}+12q^{12}-266q^{11}-113q^{10}-15q^{9}+67q^{8}+129q^{7}+110q^{6}+121q^{5}-133q^{4}-103q^{3}-94q^{2}-36q+25+78q^{-1}+135q^{-2}-13q^{-3}-19q^{-4}-58q^{-5}-49q^{-6}-38q^{-7}+8q^{-8}+71q^{-9}+13q^{-10}+19q^{-11}-8q^{-12}-15q^{-13}-29q^{-14}-14q^{-15}+19q^{-16}+2q^{-17}+11q^{-18}+4q^{-19}+3q^{-20}-9q^{-21}-7q^{-22}+4q^{-23}-2q^{-24}+2q^{-25}+q^{-26}+2q^{-27}-q^{-28}-2q^{-29}+q^{-30}$
7	$-q^{210}+2q^{209}-q^{208}-q^{207}+2q^{206}-q^{205}+q^{204}-q^{203}-2q^{202}+6q^{201}-5q^{200}-3q^{199}+5q^{198}-4q^{197}+5q^{196}+q^{195}-2q^{194}+13q^{193}-17q^{192}-12q^{191}+7q^{190}-7q^{189}+18q^{188}+9q^{187}+6q^{186}+24q^{185}-40q^{184}-38q^{183}-q^{182}-16q^{181}+48q^{180}+38q^{179}+26q^{178}+43q^{177}-80q^{176}-89q^{175}-39q^{174}-29q^{173}+115q^{172}+113q^{171}+64q^{170}+49q^{169}-172q^{168}-205q^{167}-109q^{166}-16q^{165}+291q^{164}+307q^{163}+161q^{162}-16q^{161}-454q^{160}-521q^{159}-270q^{158}+94q^{157}+726q^{156}+830q^{155}+451q^{154}-162q^{153}-1113q^{152}-1304q^{151}-738q^{150}+222q^{149}+1559q^{148}+1928q^{147}+1239q^{146}-185q^{145}-2093q^{144}-2729q^{143}-1871q^{142}+23q^{141}+2534q^{140}+3585q^{139}+2730q^{138}+379q^{137}-2886q^{136}-4463q^{135}-3683q^{134}-935q^{133}+3000q^{132}+5166q^{131}+4652q^{130}+1697q^{129}-2859q^{128}-5677q^{127}-5533q^{126}-2501q^{125}+2526q^{124}+5901q^{123}+6162q^{122}+3261q^{121}-2021q^{120}-5841q^{119}-6578q^{118}-3889q^{117}+1516q^{116}+5635q^{115}+6687q^{114}+4265q^{113}-1042q^{112}-5271q^{111}-6611q^{110}-4495q^{109}+705q^{108}+4953q^{107}+6390q^{106}+4489q^{105}-469q^{104}-4589q^{103}-6136q^{102}-4461q^{101}+335q^{100}+4346q^{99}+5844q^{98}+4336q^{97}-184q^{96}-4049q^{95}-5621q^{94}-4306q^{93}+62q^{92}+3797q^{91}+5348q^{90}+4260q^{89}+198q^{88}-3430q^{87}-5137q^{86}-4318q^{85}-473q^{84}+3046q^{83}+4820q^{82}+4332q^{81}+887q^{80}-2499q^{79}-4495q^{78}-4406q^{77}-1303q^{76}+1945q^{75}+4033q^{74}+4347q^{73}+1766q^{72}-1225q^{71}-3499q^{70}-4289q^{69}-2176q^{68}+559q^{67}+2853q^{66}+4003q^{65}+2500q^{64}+193q^{63}-2103q^{62}-3659q^{61}-2703q^{60}-801q^{59}+1334q^{58}+3083q^{57}+2678q^{56}+1345q^{55}-519q^{54}-2433q^{53}-2499q^{52}-1659q^{51}-152q^{50}+1671q^{49}+2072q^{48}+1755q^{47}+727q^{46}-940q^{45}-1537q^{44}-1615q^{43}-1031q^{42}+302q^{41}+913q^{40}+1275q^{39}+1114q^{38}+143q^{37}-345q^{36}-807q^{35}-963q^{34}-378q^{33}-83q^{32}+347q^{31}+661q^{30}+353q^{29}+327q^{28}+48q^{27}-307q^{26}-204q^{25}-356q^{24}-254q^{23}+4q^{22}-43q^{21}+226q^{20}+308q^{19}+179q^{18}+251q^{17}-33q^{16}-215q^{15}-208q^{14}-365q^{13}-148q^{12}+45q^{11}+132q^{10}+370q^{9}+259q^{8}+93q^{7}-10q^{6}-268q^{5}-257q^{4}-185q^{3}-124q^{2}+145q+207+194q^{-1}+171q^{-2}-32q^{-3}-96q^{-4}-140q^{-5}-188q^{-6}-54q^{-7}+30q^{-8}+86q^{-9}+140q^{-10}+56q^{-11}+27q^{-12}-9q^{-13}-91q^{-14}-63q^{-15}-43q^{-16}-5q^{-17}+48q^{-18}+26q^{-19}+28q^{-20}+29q^{-21}-10q^{-22}-18q^{-23}-25q^{-24}-18q^{-25}+10q^{-26}+4q^{-28}+13q^{-29}+4q^{-30}+2q^{-31}-6q^{-32}-7q^{-33}+2q^{-34}-2q^{-36}+2q^{-37}+q^{-38}+2q^{-39}-q^{-40}-2q^{-41}+q^{-42}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

9_10 9_12