9 21

9_20

9_22

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 21's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 21 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 X17,7,18,6 X7,17,8,16 X15,9,16,8
Gauss code	-1, 4, -3, 1, -6, 7, -8, 9, -2, 3, -4, 2, -5, 6, -9, 8, -7, 5
Dowker-Thistlethwaite code	4 10 14 16 12 2 18 8 6
Conway Notation	[31122]

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

[edit Notes on presentations of 9 21]

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 21"];

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 X17,7,18,6 X7,17,8,16 X15,9,16,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 12 2 18 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[31122]

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]=

${\textrm {BR}}(5,\{1,1,2,-1,2,-3,2,4,-3,4\})$

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]=

{ 5, 10, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 10.1833 A-Polynomial See Data:9 21/A-polynomial

[edit Notes for 9 21's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for 9 21's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+11t-17+11t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, 2 }
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+6q^{5}-7q^{4}+8q^{3}-6q^{2}+5q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{4}a^{-4}+2z^{2}a^{-6}+z^{2}+a^{-2}+a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-4}+z^{8}a^{-6}+3z^{7}a^{-3}+5z^{7}a^{-5}+2z^{7}a^{-7}+4z^{6}a^{-2}+4z^{6}a^{-4}+2z^{6}a^{-6}+2z^{6}a^{-8}+3z^{5}a^{-1}-3z^{5}a^{-3}-10z^{5}a^{-5}-3z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-2}-9z^{4}a^{-4}-7z^{4}a^{-6}-5z^{4}a^{-8}+z^{4}-4z^{3}a^{-1}+2z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-9}+3z^{2}a^{-2}+6z^{2}a^{-4}+5z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}+2za^{-9}-a^{-2}-a^{-6}-a^{-8}$
The A2 invariant	$q^{4}-q^{2}-1+2q^{-2}-q^{-4}+2q^{-6}+q^{-8}+q^{-12}-q^{-14}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+4q^{10}-q^{8}-4q^{6}+13q^{4}-17q^{2}+22-19q^{-2}+5q^{-4}+10q^{-6}-27q^{-8}+38q^{-10}-39q^{-12}+28q^{-14}-7q^{-16}-17q^{-18}+36q^{-20}-40q^{-22}+30q^{-24}-9q^{-26}-13q^{-28}+24q^{-30}-21q^{-32}+8q^{-34}+18q^{-36}-33q^{-38}+40q^{-40}-24q^{-42}-4q^{-44}+36q^{-46}-58q^{-48}+63q^{-50}-46q^{-52}+17q^{-54}+18q^{-56}-45q^{-58}+57q^{-60}-50q^{-62}+27q^{-64}-25q^{-68}+32q^{-70}-23q^{-72}+7q^{-74}+16q^{-76}-28q^{-78}+26q^{-80}-10q^{-82}-14q^{-84}+36q^{-86}-44q^{-88}+36q^{-90}-17q^{-92}-8q^{-94}+27q^{-96}-37q^{-98}+35q^{-100}-23q^{-102}+6q^{-104}+6q^{-106}-16q^{-108}+16q^{-110}-14q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 9_21.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+2q^{-1}-q^{-3}+2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-2q^{-13}+q^{-15}-q^{-17}$
2	$q^{10}-2q^{8}-q^{6}+6q^{4}-4q^{2}-5+11q^{-2}-3q^{-4}-9q^{-6}+11q^{-8}+q^{-10}-8q^{-12}+5q^{-14}+4q^{-16}-2q^{-18}-5q^{-20}+6q^{-22}+5q^{-24}-11q^{-26}+3q^{-28}+8q^{-30}-11q^{-32}-q^{-34}+8q^{-36}-5q^{-38}-2q^{-40}+4q^{-42}-q^{-44}-q^{-46}+q^{-48}$
3	$q^{21}-2q^{19}-q^{17}+3q^{15}+3q^{13}-4q^{11}-8q^{9}+8q^{7}+13q^{5}-9q^{3}-21q+7q^{-1}+32q^{-3}-3q^{-5}-40q^{-7}-5q^{-9}+46q^{-11}+14q^{-13}-41q^{-15}-22q^{-17}+38q^{-19}+26q^{-21}-25q^{-23}-28q^{-25}+13q^{-27}+25q^{-29}+2q^{-31}-21q^{-33}-16q^{-35}+20q^{-37}+26q^{-39}-12q^{-41}-38q^{-43}+6q^{-45}+42q^{-47}+q^{-49}-46q^{-51}-10q^{-53}+41q^{-55}+19q^{-57}-36q^{-59}-24q^{-61}+24q^{-63}+27q^{-65}-13q^{-67}-23q^{-69}+5q^{-71}+17q^{-73}+q^{-75}-11q^{-77}-2q^{-79}+6q^{-81}+2q^{-83}-3q^{-85}-q^{-87}+q^{-89}+q^{-91}-q^{-93}$
4	$q^{36}-2q^{34}-q^{32}+3q^{30}+3q^{26}-7q^{24}-3q^{22}+9q^{20}+q^{18}+9q^{16}-22q^{14}-16q^{12}+22q^{10}+21q^{8}+29q^{6}-50q^{4}-59q^{2}+16+62q^{-2}+97q^{-4}-54q^{-6}-135q^{-8}-51q^{-10}+79q^{-12}+191q^{-14}+6q^{-16}-168q^{-18}-144q^{-20}+28q^{-22}+231q^{-24}+92q^{-26}-118q^{-28}-179q^{-30}-51q^{-32}+173q^{-34}+128q^{-36}-27q^{-38}-136q^{-40}-97q^{-42}+68q^{-44}+109q^{-46}+52q^{-48}-59q^{-50}-104q^{-52}-38q^{-54}+77q^{-56}+114q^{-58}+5q^{-60}-107q^{-62}-126q^{-64}+47q^{-66}+162q^{-68}+69q^{-70}-94q^{-72}-198q^{-74}-2q^{-76}+178q^{-78}+135q^{-80}-42q^{-82}-224q^{-84}-75q^{-86}+123q^{-88}+168q^{-90}+50q^{-92}-170q^{-94}-123q^{-96}+20q^{-98}+125q^{-100}+110q^{-102}-65q^{-104}-96q^{-106}-50q^{-108}+42q^{-110}+91q^{-112}+4q^{-114}-33q^{-116}-47q^{-118}-8q^{-120}+39q^{-122}+13q^{-124}+q^{-126}-19q^{-128}-11q^{-130}+11q^{-132}+3q^{-134}+4q^{-136}-4q^{-138}-4q^{-140}+3q^{-142}+q^{-146}-q^{-148}-q^{-150}+q^{-152}$
5	$q^{55}-2q^{53}-q^{51}+3q^{49}-2q^{41}-2q^{39}+5q^{37}+4q^{35}-6q^{33}-8q^{31}-5q^{29}+6q^{27}+18q^{25}+24q^{23}-3q^{21}-46q^{19}-51q^{17}-10q^{15}+62q^{13}+109q^{11}+65q^{9}-75q^{7}-195q^{5}-154q^{3}+44q+271q^{-1}+307q^{-3}+54q^{-5}-326q^{-7}-489q^{-9}-214q^{-11}+313q^{-13}+652q^{-15}+448q^{-17}-214q^{-19}-771q^{-21}-680q^{-23}+38q^{-25}+782q^{-27}+884q^{-29}+188q^{-31}-703q^{-33}-984q^{-35}-407q^{-37}+529q^{-39}+991q^{-41}+568q^{-43}-320q^{-45}-875q^{-47}-655q^{-49}+101q^{-51}+710q^{-53}+657q^{-55}+70q^{-57}-492q^{-59}-598q^{-61}-214q^{-63}+299q^{-65}+509q^{-67}+303q^{-69}-122q^{-71}-417q^{-73}-373q^{-75}-36q^{-77}+353q^{-79}+446q^{-81}+147q^{-83}-302q^{-85}-520q^{-87}-281q^{-89}+266q^{-91}+622q^{-93}+398q^{-95}-228q^{-97}-701q^{-99}-552q^{-101}+156q^{-103}+774q^{-105}+703q^{-107}-39q^{-109}-782q^{-111}-844q^{-113}-126q^{-115}+722q^{-117}+932q^{-119}+326q^{-121}-568q^{-123}-957q^{-125}-506q^{-127}+348q^{-129}+865q^{-131}+645q^{-133}-88q^{-135}-699q^{-137}-689q^{-139}-133q^{-141}+458q^{-143}+632q^{-145}+303q^{-147}-218q^{-149}-501q^{-151}-364q^{-153}+17q^{-155}+324q^{-157}+340q^{-159}+111q^{-161}-160q^{-163}-263q^{-165}-154q^{-167}+40q^{-169}+161q^{-171}+142q^{-173}+28q^{-175}-80q^{-177}-102q^{-179}-46q^{-181}+26q^{-183}+60q^{-185}+38q^{-187}-q^{-189}-27q^{-191}-27q^{-193}-5q^{-195}+13q^{-197}+12q^{-199}+3q^{-201}-q^{-203}-6q^{-205}-4q^{-207}+3q^{-209}+2q^{-211}-q^{-213}-q^{-219}+q^{-221}+q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+2q^{-2}-q^{-4}+2q^{-6}+q^{-8}+q^{-12}-q^{-14}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
1,1	$q^{12}-4q^{10}+10q^{8}-20q^{6}+34q^{4}-52q^{2}+78-104q^{-2}+124q^{-4}-142q^{-6}+152q^{-8}-140q^{-10}+113q^{-12}-60q^{-14}+2q^{-16}+74q^{-18}-150q^{-20}+220q^{-22}-268q^{-24}+298q^{-26}-297q^{-28}+272q^{-30}-228q^{-32}+162q^{-34}-89q^{-36}+14q^{-38}+50q^{-40}-100q^{-42}+136q^{-44}-152q^{-46}+144q^{-48}-130q^{-50}+106q^{-52}-82q^{-54}+56q^{-56}-36q^{-58}+23q^{-60}-12q^{-62}+6q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{12}-q^{10}-2q^{8}+q^{6}+4q^{4}+q^{2}-7-q^{-2}+8q^{-4}-7q^{-8}+2q^{-10}+8q^{-12}-4q^{-16}+2q^{-18}+4q^{-20}-3q^{-22}+q^{-24}+2q^{-26}-2q^{-28}+2q^{-30}+7q^{-32}-q^{-34}-6q^{-36}+q^{-38}+5q^{-40}-4q^{-42}-9q^{-44}+q^{-46}+5q^{-48}-q^{-50}-4q^{-52}+3q^{-56}-2q^{-60}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-2q^{6}+4q^{2}-5+q^{-2}+8q^{-4}-8q^{-6}-q^{-8}+9q^{-10}-6q^{-12}-2q^{-14}+8q^{-16}+2q^{-22}+5q^{-24}-q^{-26}-6q^{-28}+6q^{-30}+q^{-32}-10q^{-34}+6q^{-36}+2q^{-38}-9q^{-40}+3q^{-42}+q^{-44}-4q^{-46}+2q^{-48}+q^{-50}-q^{-52}+q^{-54}$
1,0,0	$q^{5}-q^{3}-q^{-1}+2q^{-3}-q^{-5}+2q^{-7}+q^{-9}+q^{-11}+q^{-17}-q^{-19}+2q^{-21}+q^{-25}-q^{-27}+q^{-29}-q^{-31}-q^{-33}-q^{-35}$

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-2q^{6}+4q^{4}-6q^{2}+7-9q^{-2}+10q^{-4}-8q^{-6}+7q^{-8}-3q^{-10}+6q^{-14}-10q^{-16}+14q^{-18}-16q^{-20}+18q^{-22}-17q^{-24}+15q^{-26}-10q^{-28}+6q^{-30}-q^{-32}-2q^{-34}+6q^{-36}-8q^{-38}+9q^{-40}-9q^{-42}+7q^{-44}-6q^{-46}+4q^{-48}-3q^{-50}+q^{-52}-q^{-54}$
1,0	$q^{14}-2q^{10}-2q^{8}+2q^{6}+5q^{4}-6-4q^{-2}+6q^{-4}+9q^{-6}-2q^{-8}-10q^{-10}-4q^{-12}+8q^{-14}+7q^{-16}-4q^{-18}-8q^{-20}+2q^{-22}+8q^{-24}+2q^{-26}-6q^{-28}-q^{-30}+7q^{-32}+5q^{-34}-4q^{-36}-4q^{-38}+4q^{-40}+6q^{-42}-3q^{-44}-7q^{-46}+q^{-48}+8q^{-50}+q^{-52}-9q^{-54}-7q^{-56}+6q^{-58}+9q^{-60}-2q^{-62}-10q^{-64}-4q^{-66}+6q^{-68}+5q^{-70}-2q^{-72}-5q^{-74}-q^{-76}+3q^{-78}+2q^{-80}-q^{-82}-q^{-84}+q^{-88}$

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-2q^{16}+4q^{14}-6q^{12}+4q^{10}-q^{8}-4q^{6}+13q^{4}-17q^{2}+22-19q^{-2}+5q^{-4}+10q^{-6}-27q^{-8}+38q^{-10}-39q^{-12}+28q^{-14}-7q^{-16}-17q^{-18}+36q^{-20}-40q^{-22}+30q^{-24}-9q^{-26}-13q^{-28}+24q^{-30}-21q^{-32}+8q^{-34}+18q^{-36}-33q^{-38}+40q^{-40}-24q^{-42}-4q^{-44}+36q^{-46}-58q^{-48}+63q^{-50}-46q^{-52}+17q^{-54}+18q^{-56}-45q^{-58}+57q^{-60}-50q^{-62}+27q^{-64}-25q^{-68}+32q^{-70}-23q^{-72}+7q^{-74}+16q^{-76}-28q^{-78}+26q^{-80}-10q^{-82}-14q^{-84}+36q^{-86}-44q^{-88}+36q^{-90}-17q^{-92}-8q^{-94}+27q^{-96}-37q^{-98}+35q^{-100}-23q^{-102}+6q^{-104}+6q^{-106}-16q^{-108}+16q^{-110}-14q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

.

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 21"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+11t-17+11t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{4}+3z^{2}+1$

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]=

{ 43, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-4}+z^{8}a^{-6}+3z^{7}a^{-3}+5z^{7}a^{-5}+2z^{7}a^{-7}+4z^{6}a^{-2}+4z^{6}a^{-4}+2z^{6}a^{-6}+2z^{6}a^{-8}+3z^{5}a^{-1}-3z^{5}a^{-3}-10z^{5}a^{-5}-3z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-2}-9z^{4}a^{-4}-7z^{4}a^{-6}-5z^{4}a^{-8}+z^{4}-4z^{3}a^{-1}+2z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-9}+3z^{2}a^{-2}+6z^{2}a^{-4}+5z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}+2za^{-9}-a^{-2}-a^{-6}-a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n129,}

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 21"];

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]=

{ $-2t^{2}+11t-17+11t^{-1}-2t^{-2}$, $-q^{8}+2q^{7}-4q^{6}+6q^{5}-7q^{4}+8q^{3}-6q^{2}+5q-3+q^{-1}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{K11n129,}

Vassiliev invariants

V2 and V3:

(3, 6)

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 $12$ $48$ $72$ $238$ $50$ $576$ $1248$ $192$ $272$ $288$ $1152$ $2856$ $600$ ${\frac {65311}{10}}$ $-{\frac {3182}{5}}$ ${\frac {18154}{5}}$ ${\frac {235}{2}}$ ${\frac {5631}{10}}$

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=$2 is the signature of 9 21. 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 17                   1 -1 15                 1   1 13               3 1   -2 11             3 1     2 9           4 3       -1 7         4 3         1 5       2 4           2 3     3 4             -1 1   1 3               2 -1   2                 -2 -3 1                   1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{23}-2q^{22}+6q^{20}-8q^{19}-3q^{18}+19q^{17}-17q^{16}-13q^{15}+38q^{14}-22q^{13}-27q^{12}+54q^{11}-21q^{10}-38q^{9}+57q^{8}-15q^{7}-37q^{6}+44q^{5}-6q^{4}-27q^{3}+24q^{2}-13+8q^{-1}+q^{-2}-3q^{-3}+q^{-4}$
3	$-q^{45}+2q^{44}-2q^{42}-3q^{41}+7q^{40}+4q^{39}-10q^{38}-12q^{37}+19q^{36}+20q^{35}-22q^{34}-40q^{33}+29q^{32}+60q^{31}-25q^{30}-88q^{29}+17q^{28}+115q^{27}-3q^{26}-139q^{25}-19q^{24}+162q^{23}+38q^{22}-175q^{21}-63q^{20}+188q^{19}+76q^{18}-181q^{17}-99q^{16}+183q^{15}+99q^{14}-158q^{13}-111q^{12}+142q^{11}+102q^{10}-107q^{9}-99q^{8}+82q^{7}+83q^{6}-52q^{5}-67q^{4}+31q^{3}+48q^{2}-15q-32+6q^{-1}+20q^{-2}-3q^{-3}-10q^{-4}+q^{-5}+4q^{-6}+q^{-7}-3q^{-8}+q^{-9}$
4	$q^{74}-2q^{73}+2q^{71}-q^{70}+4q^{69}-9q^{68}+10q^{66}-2q^{65}+12q^{64}-31q^{63}-8q^{62}+30q^{61}+10q^{60}+38q^{59}-78q^{58}-47q^{57}+44q^{56}+47q^{55}+125q^{54}-127q^{53}-139q^{52}-2q^{51}+78q^{50}+300q^{49}-112q^{48}-244q^{47}-145q^{46}+31q^{45}+520q^{44}+6q^{43}-289q^{42}-343q^{41}-118q^{40}+702q^{39}+183q^{38}-246q^{37}-523q^{36}-314q^{35}+806q^{34}+346q^{33}-153q^{32}-638q^{31}-487q^{30}+825q^{29}+458q^{28}-44q^{27}-675q^{26}-602q^{25}+759q^{24}+503q^{23}+67q^{22}-618q^{21}-643q^{20}+594q^{19}+464q^{18}+176q^{17}-463q^{16}-598q^{15}+370q^{14}+336q^{13}+237q^{12}-253q^{11}-459q^{10}+167q^{9}+164q^{8}+213q^{7}-79q^{6}-274q^{5}+55q^{4}+34q^{3}+129q^{2}+2q-123+20q^{-1}-12q^{-2}+54q^{-3}+11q^{-4}-44q^{-5}+12q^{-6}-11q^{-7}+16q^{-8}+5q^{-9}-13q^{-10}+4q^{-11}-3q^{-12}+4q^{-13}+q^{-14}-3q^{-15}+q^{-16}$
5	$-q^{110}+2q^{109}-2q^{107}+q^{106}-2q^{104}+5q^{103}+q^{102}-9q^{101}-q^{100}+5q^{99}+2q^{98}+14q^{97}+2q^{96}-27q^{95}-23q^{94}+5q^{93}+28q^{92}+53q^{91}+24q^{90}-61q^{89}-95q^{88}-51q^{87}+50q^{86}+161q^{85}+138q^{84}-42q^{83}-216q^{82}-245q^{81}-59q^{80}+264q^{79}+409q^{78}+187q^{77}-232q^{76}-552q^{75}-440q^{74}+127q^{73}+692q^{72}+708q^{71}+97q^{70}-726q^{69}-1031q^{68}-429q^{67}+682q^{66}+1319q^{65}+830q^{64}-506q^{63}-1548q^{62}-1283q^{61}+231q^{60}+1708q^{59}+1724q^{58}+100q^{57}-1758q^{56}-2131q^{55}-487q^{54}+1770q^{53}+2467q^{52}+842q^{51}-1687q^{50}-2749q^{49}-1195q^{48}+1621q^{47}+2940q^{46}+1468q^{45}-1463q^{44}-3105q^{43}-1742q^{42}+1382q^{41}+3158q^{40}+1917q^{39}-1164q^{38}-3198q^{37}-2131q^{36}+1045q^{35}+3114q^{34}+2212q^{33}-739q^{32}-2992q^{31}-2341q^{30}+532q^{29}+2730q^{28}+2318q^{27}-177q^{26}-2405q^{25}-2288q^{24}-77q^{23}+1974q^{22}+2098q^{21}+378q^{20}-1517q^{19}-1865q^{18}-539q^{17}+1038q^{16}+1521q^{15}+659q^{14}-626q^{13}-1169q^{12}-641q^{11}+294q^{10}+803q^{9}+568q^{8}-69q^{7}-507q^{6}-437q^{5}-45q^{4}+276q^{3}+293q^{2}+94q-127-184q^{-1}-81q^{-2}+49q^{-3}+95q^{-4}+53q^{-5}-7q^{-6}-44q^{-7}-37q^{-8}+2q^{-9}+23q^{-10}+12q^{-11}-2q^{-12}-q^{-13}-10q^{-14}-4q^{-15}+11q^{-16}+q^{-17}-5q^{-18}+q^{-19}-3q^{-21}+4q^{-22}+q^{-23}-3q^{-24}+q^{-25}$
6	$q^{153}-2q^{152}+2q^{150}-q^{149}-2q^{147}+6q^{146}-6q^{145}-2q^{144}+11q^{143}-3q^{142}-4q^{141}-13q^{140}+15q^{139}-12q^{138}-2q^{137}+40q^{136}+5q^{135}-13q^{134}-55q^{133}+11q^{132}-40q^{131}+123q^{129}+70q^{128}+10q^{127}-140q^{126}-54q^{125}-175q^{124}-67q^{123}+257q^{122}+283q^{121}+212q^{120}-147q^{119}-148q^{118}-548q^{117}-432q^{116}+213q^{115}+571q^{114}+754q^{113}+268q^{112}+108q^{111}-1003q^{110}-1279q^{109}-499q^{108}+387q^{107}+1340q^{106}+1291q^{105}+1343q^{104}-779q^{103}-2139q^{102}-2047q^{101}-1012q^{100}+990q^{99}+2286q^{98}+3647q^{97}+947q^{96}-1858q^{95}-3616q^{94}-3609q^{93}-1140q^{92}+2000q^{91}+6025q^{90}+4038q^{89}+347q^{88}-3910q^{87}-6293q^{86}-4749q^{85}-209q^{84}+7174q^{83}+7295q^{82}+4009q^{81}-2419q^{80}-7843q^{79}-8568q^{78}-3712q^{77}+6702q^{76}+9565q^{75}+7832q^{74}+172q^{73}-7986q^{72}-11495q^{71}-7246q^{70}+5260q^{69}+10601q^{68}+10808q^{67}+2762q^{66}-7292q^{65}-13269q^{64}-9958q^{63}+3692q^{62}+10823q^{61}+12730q^{60}+4755q^{59}-6367q^{58}-14165q^{57}-11759q^{56}+2300q^{55}+10589q^{54}+13842q^{53}+6259q^{52}-5291q^{51}-14366q^{50}-12922q^{49}+804q^{48}+9790q^{47}+14251q^{46}+7601q^{45}-3686q^{44}-13634q^{43}-13488q^{42}-1121q^{41}+7974q^{40}+13616q^{39}+8730q^{38}-1279q^{37}-11490q^{36}-13007q^{35}-3237q^{34}+4967q^{33}+11417q^{32}+9006q^{31}+1493q^{30}-7898q^{29}-10923q^{28}-4636q^{27}+1467q^{26}+7737q^{25}+7748q^{24}+3485q^{23}-3817q^{22}-7419q^{21}-4482q^{20}-1153q^{19}+3722q^{18}+5140q^{17}+3793q^{16}-754q^{15}-3722q^{14}-2957q^{13}-2008q^{12}+860q^{11}+2376q^{10}+2690q^{9}+525q^{8}-1187q^{7}-1203q^{6}-1489q^{5}-297q^{4}+600q^{3}+1306q^{2}+522q-136-164q^{-1}-670q^{-2}-356q^{-3}-44q^{-4}+448q^{-5}+191q^{-6}+44q^{-7}+128q^{-8}-191q^{-9}-149q^{-10}-105q^{-11}+126q^{-12}+19q^{-13}+3q^{-14}+97q^{-15}-35q^{-16}-35q^{-17}-46q^{-18}+42q^{-19}-11q^{-20}-13q^{-21}+34q^{-22}-7q^{-23}-4q^{-24}-14q^{-25}+18q^{-26}-4q^{-27}-9q^{-28}+9q^{-29}-3q^{-30}-3q^{-32}+4q^{-33}+q^{-34}-3q^{-35}+q^{-36}$

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.

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