9 15

9_14

9_16

Visit 9 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 15's page at Knotilus! Visit 9 15's page at the original Knot Atlas!

9 15 Quick Notes

9 15 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12
Gauss code	-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 14 10 2 18 16 6 12
Conway Notation	[2322]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 9.8855 A-Polynomial See Data:9 15/A-polynomial

[edit Notes for 9 15'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 15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+10t-15+10t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 39, 2 }
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+6q^{5}-6q^{4}+7q^{3}-6q^{2}+4q-2+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{4}a^{-4}-z^{2}a^{-2}+2z^{2}a^{-6}+z^{2}-a^{-2}+a^{-4}+a^{-6}-a^{-8}+1$
Kauffman polynomial (db, data sources)	$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+4z^{7}a^{-5}+2z^{7}a^{-7}+2z^{6}a^{-2}+z^{6}a^{-4}+z^{6}a^{-6}+2z^{6}a^{-8}+2z^{5}a^{-1}-z^{5}a^{-3}-7z^{5}a^{-5}-3z^{5}a^{-7}+z^{5}a^{-9}-4z^{4}a^{-6}-5z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-5}-z^{3}a^{-7}-3z^{3}a^{-9}-3z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}a^{-6}+3z^{2}a^{-8}-2z^{2}+za^{-1}+za^{-3}-za^{-5}+za^{-7}+2za^{-9}+a^{-2}+a^{-4}-a^{-6}-a^{-8}+1$
The A2 invariant	$q^{4}+2q^{-2}-2q^{-4}+2q^{-12}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
The G2 invariant	$q^{18}-q^{16}+3q^{14}-3q^{12}+2q^{10}-3q^{6}+8q^{4}-9q^{2}+11-9q^{-2}+5q^{-4}+4q^{-6}-13q^{-8}+23q^{-10}-26q^{-12}+21q^{-14}-13q^{-16}-5q^{-18}+20q^{-20}-31q^{-22}+33q^{-24}-20q^{-26}+2q^{-28}+14q^{-30}-23q^{-32}+15q^{-34}-2q^{-36}-13q^{-38}+24q^{-40}-23q^{-42}+9q^{-44}+17q^{-46}-34q^{-48}+46q^{-50}-42q^{-52}+21q^{-54}+6q^{-56}-28q^{-58}+43q^{-60}-44q^{-62}+36q^{-64}-11q^{-66}-11q^{-68}+26q^{-70}-30q^{-72}+20q^{-74}-q^{-76}-15q^{-78}+21q^{-80}-15q^{-82}+3q^{-84}+18q^{-86}-31q^{-88}+33q^{-90}-23q^{-92}+18q^{-96}-32q^{-98}+33q^{-100}-24q^{-102}+10q^{-104}+3q^{-106}-15q^{-108}+17q^{-110}-15q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 9_15.

A1 Invariants.

Weight	Invariant
1	$q^{3}-q+2q^{-1}-2q^{-3}+q^{-5}+q^{-7}+2q^{-11}-2q^{-13}+q^{-15}-q^{-17}$
2	$q^{10}-q^{8}-q^{6}+3q^{4}-3q^{2}-2+9q^{-2}-4q^{-4}-6q^{-6}+11q^{-8}-q^{-10}-7q^{-12}+5q^{-14}+2q^{-16}-3q^{-18}-3q^{-20}+5q^{-22}+2q^{-24}-8q^{-26}+5q^{-28}+6q^{-30}-10q^{-32}+q^{-34}+7q^{-36}-6q^{-38}-2q^{-40}+4q^{-42}-q^{-44}-q^{-46}+q^{-48}$
3	$q^{21}-q^{19}-q^{17}+2q^{13}-q^{11}-2q^{9}+4q^{7}+4q^{5}-7q^{3}-8q+11q^{-1}+16q^{-3}-14q^{-5}-25q^{-7}+13q^{-9}+33q^{-11}-5q^{-13}-38q^{-15}+38q^{-19}+8q^{-21}-28q^{-23}-14q^{-25}+21q^{-27}+16q^{-29}-9q^{-31}-19q^{-33}-2q^{-35}+17q^{-37}+11q^{-39}-19q^{-41}-21q^{-43}+19q^{-45}+27q^{-47}-14q^{-49}-33q^{-51}+10q^{-53}+36q^{-55}-37q^{-59}-7q^{-61}+28q^{-63}+15q^{-65}-21q^{-67}-18q^{-69}+12q^{-71}+16q^{-73}-3q^{-75}-12q^{-77}-q^{-79}+7q^{-81}+2q^{-83}-3q^{-85}-q^{-87}+q^{-89}+q^{-91}-q^{-93}$
4	$q^{36}-q^{34}-q^{32}-q^{28}+4q^{26}-q^{24}+2q^{20}-5q^{18}+3q^{16}-7q^{14}+2q^{12}+15q^{10}-q^{8}-2q^{6}-32q^{4}-9q^{2}+41+34q^{-2}+13q^{-4}-78q^{-6}-62q^{-8}+46q^{-10}+92q^{-12}+74q^{-14}-95q^{-16}-135q^{-18}-7q^{-20}+113q^{-22}+151q^{-24}-49q^{-26}-159q^{-28}-75q^{-30}+68q^{-32}+162q^{-34}+20q^{-36}-103q^{-38}-97q^{-40}-2q^{-42}+107q^{-44}+62q^{-46}-25q^{-48}-78q^{-50}-50q^{-52}+33q^{-54}+76q^{-56}+40q^{-58}-56q^{-60}-85q^{-62}-25q^{-64}+91q^{-66}+95q^{-68}-35q^{-70}-115q^{-72}-81q^{-74}+96q^{-76}+143q^{-78}+5q^{-80}-118q^{-82}-136q^{-84}+55q^{-86}+152q^{-88}+67q^{-90}-67q^{-92}-161q^{-94}-20q^{-96}+101q^{-98}+99q^{-100}+15q^{-102}-113q^{-104}-64q^{-106}+17q^{-108}+70q^{-110}+61q^{-112}-37q^{-114}-47q^{-116}-28q^{-118}+16q^{-120}+44q^{-122}+4q^{-124}-9q^{-126}-21q^{-128}-7q^{-130}+14q^{-132}+4q^{-134}+3q^{-136}-5q^{-138}-4q^{-140}+3q^{-142}+q^{-146}-q^{-148}-q^{-150}+q^{-152}$
5	$q^{55}-q^{53}-q^{51}-q^{47}+q^{45}+4q^{43}+q^{41}-2q^{39}-5q^{35}-5q^{33}+3q^{31}+6q^{29}+7q^{27}+6q^{25}-5q^{23}-20q^{21}-19q^{19}+2q^{17}+30q^{15}+45q^{13}+21q^{11}-37q^{9}-88q^{7}-68q^{5}+34q^{3}+137q+146q^{-1}+8q^{-3}-182q^{-5}-260q^{-7}-97q^{-9}+206q^{-11}+382q^{-13}+235q^{-15}-169q^{-17}-487q^{-19}-410q^{-21}+67q^{-23}+553q^{-25}+581q^{-27}+84q^{-29}-529q^{-31}-704q^{-33}-266q^{-35}+436q^{-37}+765q^{-39}+412q^{-41}-287q^{-43}-721q^{-45}-510q^{-47}+116q^{-49}+604q^{-51}+547q^{-53}+28q^{-55}-451q^{-57}-501q^{-59}-150q^{-61}+277q^{-63}+434q^{-65}+222q^{-67}-136q^{-69}-336q^{-71}-267q^{-73}+5q^{-75}+263q^{-77}+298q^{-79}+91q^{-81}-198q^{-83}-334q^{-85}-181q^{-87}+156q^{-89}+385q^{-91}+269q^{-93}-125q^{-95}-443q^{-97}-366q^{-99}+79q^{-101}+496q^{-103}+476q^{-105}-16q^{-107}-530q^{-109}-576q^{-111}-88q^{-113}+522q^{-115}+667q^{-117}+208q^{-119}-441q^{-121}-711q^{-123}-352q^{-125}+317q^{-127}+695q^{-129}+462q^{-131}-137q^{-133}-595q^{-135}-548q^{-137}-43q^{-139}+449q^{-141}+534q^{-143}+195q^{-145}-252q^{-147}-461q^{-149}-292q^{-151}+79q^{-153}+328q^{-155}+302q^{-157}+61q^{-159}-178q^{-161}-252q^{-163}-137q^{-165}+57q^{-167}+168q^{-169}+141q^{-171}+26q^{-173}-80q^{-175}-110q^{-177}-59q^{-179}+21q^{-181}+63q^{-183}+54q^{-185}+9q^{-187}-25q^{-189}-34q^{-191}-19q^{-193}+7q^{-195}+18q^{-197}+11q^{-199}-4q^{-203}-7q^{-205}-3q^{-207}+4q^{-209}+2q^{-211}-q^{-213}-q^{-219}+q^{-221}+q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}+2q^{-2}-2q^{-4}+2q^{-12}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
1,1	$q^{12}-2q^{10}+6q^{8}-10q^{6}+17q^{4}-26q^{2}+36-52q^{-2}+68q^{-4}-82q^{-6}+98q^{-8}-102q^{-10}+106q^{-12}-82q^{-14}+52q^{-16}-8q^{-18}-47q^{-20}+102q^{-22}-156q^{-24}+194q^{-26}-217q^{-28}+220q^{-30}-204q^{-32}+174q^{-34}-126q^{-36}+72q^{-38}-12q^{-40}-38q^{-42}+78q^{-44}-108q^{-46}+118q^{-48}-116q^{-50}+98q^{-52}-80q^{-54}+58q^{-56}-38q^{-58}+23q^{-60}-12q^{-62}+6q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{12}-q^{8}+2q^{4}-4+7q^{-4}-q^{-6}-6q^{-8}+3q^{-10}+7q^{-12}+q^{-14}-4q^{-16}+3q^{-18}+3q^{-20}-3q^{-22}-q^{-24}-3q^{-28}-q^{-30}+5q^{-32}-q^{-34}-2q^{-36}+3q^{-38}+6q^{-40}-2q^{-42}-6q^{-44}+2q^{-46}+3q^{-48}-3q^{-50}-5q^{-52}+3q^{-56}-2q^{-60}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-q^{6}+q^{4}+3q^{2}-3+6q^{-4}-6q^{-6}-2q^{-8}+9q^{-10}-5q^{-12}-2q^{-14}+6q^{-16}-2q^{-18}-2q^{-20}+q^{-22}+4q^{-24}-2q^{-28}+6q^{-30}+3q^{-32}-8q^{-34}+4q^{-36}+2q^{-38}-9q^{-40}+2q^{-42}+q^{-44}-4q^{-46}+2q^{-48}+q^{-50}-q^{-52}+q^{-54}$
1,0,0	$q^{5}+q+2q^{-3}-2q^{-5}-q^{-9}+q^{-15}+2q^{-17}+2q^{-21}+q^{-25}-q^{-27}+q^{-29}-q^{-31}-q^{-33}-q^{-35}$

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-q^{16}+3q^{14}-3q^{12}+2q^{10}-3q^{6}+8q^{4}-9q^{2}+11-9q^{-2}+5q^{-4}+4q^{-6}-13q^{-8}+23q^{-10}-26q^{-12}+21q^{-14}-13q^{-16}-5q^{-18}+20q^{-20}-31q^{-22}+33q^{-24}-20q^{-26}+2q^{-28}+14q^{-30}-23q^{-32}+15q^{-34}-2q^{-36}-13q^{-38}+24q^{-40}-23q^{-42}+9q^{-44}+17q^{-46}-34q^{-48}+46q^{-50}-42q^{-52}+21q^{-54}+6q^{-56}-28q^{-58}+43q^{-60}-44q^{-62}+36q^{-64}-11q^{-66}-11q^{-68}+26q^{-70}-30q^{-72}+20q^{-74}-q^{-76}-15q^{-78}+21q^{-80}-15q^{-82}+3q^{-84}+18q^{-86}-31q^{-88}+33q^{-90}-23q^{-92}+18q^{-96}-32q^{-98}+33q^{-100}-24q^{-102}+10q^{-104}+3q^{-106}-15q^{-108}+17q^{-110}-15q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

.

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 15"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+10t-15+10t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 39, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{-2}-z^{4}a^{-4}-z^{2}a^{-2}+2z^{2}a^{-6}+z^{2}-a^{-2}+a^{-4}+a^{-6}-a^{-8}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(2, 5)

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 $8$ $40$ $32$ ${\frac {508}{3}}$ ${\frac {92}{3}}$ $320$ ${\frac {2416}{3}}$ ${\frac {352}{3}}$ $168$ ${\frac {256}{3}}$ $800$ ${\frac {4064}{3}}$ ${\frac {736}{3}}$ ${\frac {59071}{15}}$ $-{\frac {1628}{5}}$ ${\frac {92164}{45}}$ ${\frac {881}{9}}$ ${\frac {4351}{15}}$

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=$2 is the signature of 9 15. 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						3	3				0
7					4	3					1
5				2	3						1
3			2	4							-2
1		1	3								2
-1		1									-1
-3	1										1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 15]]
Out[2]=	9
In[3]:=	PD[Knot[9, 15]]
Out[3]=	PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[13, 17, 14, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[11, 1, 12, 18], X[17, 13, 18, 12]]
In[4]:=	GaussCode[Knot[9, 15]]
Out[4]=	GaussCode[-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8]
In[5]:=	BR[Knot[9, 15]]
Out[5]=	BR[5, {1, 1, 1, 2, -1, -3, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[9, 15]][t]
Out[6]=	2 10 2 -15 - -- + -- + 10 t - 2 t 2 t t
In[7]:=	Conway[Knot[9, 15]][z]
Out[7]=	2 4 1 + 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 15], Knot[10, 165], Knot[11, NonAlternating, 63], Knot[11, NonAlternating, 101]}
In[9]:=	{KnotDet[Knot[9, 15]], KnotSignature[Knot[9, 15]]}
Out[9]=	{39, 2}
In[10]:=	J=Jones[Knot[9, 15]][q]
Out[10]=	1 2 3 4 5 6 7 8 -2 + - + 4 q - 6 q + 7 q - 6 q + 6 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 15]}
In[12]:=	A2Invariant[Knot[9, 15]][q]
Out[12]=	-4 2 4 12 16 20 22 24 26 q + 2 q - 2 q + 2 q + 2 q - q + q - q - q
In[13]:=	Kauffman[Knot[9, 15]][a, z]
Out[13]=	2 -8 -6 -4 -2 2 z z z z z 2 3 z 1 - a - a + a + a + --- + -- - -- + -- + - - 2 z + ---- + 9 7 5 3 a 8 a a a a a 2 2 2 3 3 3 3 4 4 2 z 2 z 3 z 3 z z 5 z 3 z 4 5 z 4 z ---- - ---- - ---- - ---- - -- + ---- - ---- + z - ---- - ---- + 6 4 2 9 7 5 a 8 6 a a a a a a a a 5 5 5 5 5 6 6 6 6 7 7 z 3 z 7 z z 2 z 2 z z z 2 z 2 z 4 z -- - ---- - ---- - -- + ---- + ---- + -- + -- + ---- + ---- + ---- + 9 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a a 7 8 8 2 z z z ---- + -- + -- 3 6 4 a a a
In[14]:=	{Vassiliev[2][Knot[9, 15]], Vassiliev[3][Knot[9, 15]]}
Out[14]=	{0, 5}
In[15]:=	Kh[Knot[9, 15]][q, t]
Out[15]=	3 1 1 q 3 5 5 2 7 2 3 q + 2 q + ----- + --- + - + 4 q t + 2 q t + 3 q t + 4 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 3 q t + 3 q t + 3 q t + 3 q t + q t + 3 q t + q t + 15 6 17 7 q t + q t