9 3

9_2

9_4

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

9 3 Further Notes and Views

Knot presentations

Planar diagram presentation	X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X14,6,15,5 X16,8,17,7 X2,12,3,11 X4,14,5,13 X6,16,7,15
Gauss code	1, -7, 2, -8, 5, -9, 6, -1, 3, -4, 7, -2, 8, -5, 9, -6, 4, -3
Dowker-Thistlethwaite code	8 12 14 16 18 2 4 6 10
Conway Notation	[63]

Minimum Braid Representative: Length is 10, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [5][-16] Hyperbolic Volume 4.99486 A-Polynomial See Data:9 3/A-polynomial

[edit Notes for 9 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $3$ Topological 4 genus $3$ Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for 9 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-3t^{2}+3t-3+3t^{-1}-3t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+9z^{4}+9z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, 6 }
Jones polynomial	$-q^{12}+q^{11}-2q^{10}+3q^{9}-3q^{8}+3q^{7}-2q^{6}+2q^{5}-q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+z^{6}a^{-8}+5z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}a^{-10}+6z^{2}a^{-6}+7z^{2}a^{-8}-4z^{2}a^{-10}+a^{-6}+3a^{-8}-3a^{-10}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-7}+2z^{7}a^{-9}+z^{7}a^{-11}+z^{6}a^{-6}-5z^{6}a^{-8}-5z^{6}a^{-10}+z^{6}a^{-12}-4z^{5}a^{-7}-8z^{5}a^{-9}-3z^{5}a^{-11}+z^{5}a^{-13}-5z^{4}a^{-6}+9z^{4}a^{-8}+11z^{4}a^{-10}-2z^{4}a^{-12}+z^{4}a^{-14}+3z^{3}a^{-7}+9z^{3}a^{-9}+4z^{3}a^{-11}-z^{3}a^{-13}+z^{3}a^{-15}+6z^{2}a^{-6}-9z^{2}a^{-8}-11z^{2}a^{-10}+3z^{2}a^{-12}-z^{2}a^{-14}-4za^{-9}-za^{-11}+za^{-13}-2za^{-15}-a^{-6}+3a^{-8}+3a^{-10}$
The A2 invariant	$q^{-10}+q^{-14}+q^{-18}+q^{-20}+q^{-22}+2q^{-24}-q^{-30}-q^{-32}-q^{-34}-q^{-36}$
The G2 invariant	$q^{-50}+q^{-54}-q^{-56}+q^{-58}+3q^{-64}-2q^{-66}+3q^{-68}-q^{-70}+q^{-72}+2q^{-74}-3q^{-76}+3q^{-78}-q^{-80}+q^{-82}+q^{-84}-q^{-86}+2q^{-88}+q^{-90}+2q^{-94}-q^{-96}+q^{-98}+2q^{-100}-2q^{-102}+3q^{-104}-q^{-106}+2q^{-108}-q^{-112}+2q^{-114}-2q^{-116}+3q^{-118}-2q^{-120}+q^{-124}-2q^{-126}+q^{-128}-q^{-130}-q^{-132}+q^{-134}-q^{-136}-q^{-138}-q^{-142}-q^{-146}-2q^{-148}-q^{-152}-q^{-156}-q^{-160}-q^{-162}-q^{-164}-q^{-166}+2q^{-168}-2q^{-170}+q^{-172}+q^{-178}-q^{-180}+q^{-182}-q^{-184}+q^{-186}-q^{-190}+q^{-192}+q^{-196}$

Further Quantum Invariants

Further quantum knot invariants for 9_3.

A1 Invariants.

Weight	Invariant
1	$q^{-5}+q^{-9}+q^{-13}+q^{-19}-q^{-21}-q^{-25}$
2	$q^{-10}+2q^{-16}+q^{-18}-q^{-20}+q^{-22}+q^{-24}-q^{-26}+q^{-30}+q^{-36}-q^{-40}-q^{-48}+q^{-50}-q^{-52}-2q^{-54}-q^{-58}+q^{-62}+q^{-68}$
3	$q^{-15}+q^{-21}+2q^{-23}+q^{-25}-q^{-27}-q^{-29}+2q^{-31}+2q^{-33}-2q^{-37}+2q^{-41}+q^{-43}-q^{-45}-q^{-47}+q^{-51}+q^{-53}-q^{-57}+2q^{-61}-2q^{-65}-q^{-67}+q^{-69}-q^{-71}-2q^{-73}+q^{-77}-q^{-81}-q^{-83}-q^{-85}+q^{-87}+q^{-89}-2q^{-91}-q^{-93}-q^{-95}+2q^{-97}-q^{-101}-q^{-103}+q^{-105}+2q^{-107}+q^{-109}-q^{-111}+2q^{-115}+q^{-117}-q^{-121}-q^{-129}$
4	$q^{-20}+q^{-26}+q^{-28}+2q^{-30}-q^{-34}+4q^{-40}+2q^{-42}-q^{-44}-2q^{-46}-3q^{-48}+2q^{-50}+3q^{-52}+2q^{-54}-4q^{-58}-q^{-60}+q^{-62}+2q^{-64}+2q^{-66}-q^{-68}-q^{-72}-q^{-74}+q^{-76}+q^{-78}+3q^{-80}-4q^{-84}-3q^{-86}+4q^{-90}+2q^{-92}-4q^{-94}-4q^{-96}-q^{-98}+4q^{-100}+3q^{-102}-3q^{-104}-4q^{-106}-q^{-108}+2q^{-110}+2q^{-112}-2q^{-114}-2q^{-116}+q^{-120}+q^{-122}-2q^{-124}-2q^{-126}+q^{-130}+q^{-132}+q^{-134}-q^{-136}-3q^{-138}-2q^{-140}+2q^{-142}+6q^{-144}+3q^{-146}-q^{-148}-7q^{-150}-3q^{-152}+6q^{-154}+6q^{-156}+2q^{-158}-7q^{-160}-4q^{-162}+3q^{-164}+5q^{-166}+4q^{-168}-3q^{-170}-4q^{-172}+q^{-174}+2q^{-176}+3q^{-178}-2q^{-180}-3q^{-182}-q^{-184}+2q^{-188}-q^{-190}-q^{-192}-q^{-194}-q^{-196}+q^{-198}+q^{-208}$
5	$q^{-25}+q^{-31}+q^{-33}+q^{-35}+q^{-37}-q^{-41}+2q^{-45}+2q^{-47}+3q^{-49}+q^{-51}-2q^{-53}-4q^{-55}-2q^{-57}+q^{-59}+4q^{-61}+5q^{-63}+2q^{-65}-2q^{-67}-5q^{-69}-4q^{-71}+3q^{-75}+5q^{-77}+3q^{-79}-q^{-81}-4q^{-83}-3q^{-85}+q^{-89}+2q^{-91}+2q^{-93}+q^{-95}+q^{-97}+q^{-99}-2q^{-101}-4q^{-103}-4q^{-105}+5q^{-109}+6q^{-111}+2q^{-113}-5q^{-115}-10q^{-117}-6q^{-119}+3q^{-121}+9q^{-123}+7q^{-125}-q^{-127}-9q^{-129}-10q^{-131}-3q^{-133}+7q^{-135}+10q^{-137}+4q^{-139}-4q^{-141}-9q^{-143}-6q^{-145}+2q^{-147}+8q^{-149}+5q^{-151}-3q^{-153}-7q^{-155}-7q^{-157}+6q^{-161}+5q^{-163}-4q^{-167}-3q^{-169}+q^{-171}+3q^{-173}+3q^{-175}-2q^{-177}-3q^{-179}+2q^{-183}+2q^{-185}+q^{-187}-q^{-189}-2q^{-191}-2q^{-193}+2q^{-197}+4q^{-199}+6q^{-201}+2q^{-203}-5q^{-205}-8q^{-207}-4q^{-209}+3q^{-211}+11q^{-213}+13q^{-215}-q^{-217}-12q^{-219}-13q^{-221}-3q^{-223}+9q^{-225}+15q^{-227}+5q^{-229}-8q^{-231}-11q^{-233}-6q^{-235}+5q^{-237}+8q^{-239}+3q^{-241}-4q^{-243}-6q^{-245}-3q^{-247}+3q^{-249}+4q^{-251}+q^{-253}-4q^{-255}-3q^{-257}-q^{-259}+2q^{-261}+2q^{-263}+q^{-265}-2q^{-267}-3q^{-269}-q^{-271}+q^{-273}+q^{-275}+2q^{-277}+q^{-279}-q^{-281}+q^{-289}+q^{-291}-q^{-305}$
6	$q^{-30}+q^{-36}+q^{-38}+q^{-40}+q^{-44}-q^{-48}+q^{-50}+2q^{-52}+3q^{-54}+q^{-56}+2q^{-58}-q^{-60}-4q^{-62}-3q^{-64}-q^{-66}+3q^{-68}+3q^{-70}+7q^{-72}+4q^{-74}-2q^{-76}-5q^{-78}-7q^{-80}-4q^{-82}-2q^{-84}+6q^{-86}+8q^{-88}+6q^{-90}+2q^{-92}-3q^{-94}-6q^{-96}-9q^{-98}-2q^{-100}+2q^{-102}+6q^{-104}+6q^{-106}+3q^{-108}+q^{-110}-4q^{-112}-2q^{-114}-2q^{-116}-q^{-120}-2q^{-122}+5q^{-128}+6q^{-130}+5q^{-132}-3q^{-134}-11q^{-136}-11q^{-138}-9q^{-140}+2q^{-142}+12q^{-144}+17q^{-146}+9q^{-148}-5q^{-150}-15q^{-152}-20q^{-154}-11q^{-156}+4q^{-158}+18q^{-160}+19q^{-162}+9q^{-164}-3q^{-166}-18q^{-168}-20q^{-170}-11q^{-172}+5q^{-174}+16q^{-176}+17q^{-178}+12q^{-180}-5q^{-182}-18q^{-184}-19q^{-186}-9q^{-188}+5q^{-190}+16q^{-192}+20q^{-194}+7q^{-196}-10q^{-198}-19q^{-200}-15q^{-202}-3q^{-204}+10q^{-206}+20q^{-208}+11q^{-210}-5q^{-212}-14q^{-214}-12q^{-216}-3q^{-218}+7q^{-220}+16q^{-222}+10q^{-224}-4q^{-226}-10q^{-228}-8q^{-230}-2q^{-232}+4q^{-234}+8q^{-236}+4q^{-238}-4q^{-240}-4q^{-242}+2q^{-246}+3q^{-248}+3q^{-250}-3q^{-254}-3q^{-256}+3q^{-260}+4q^{-262}+4q^{-264}+2q^{-266}+q^{-268}-4q^{-270}-8q^{-272}-7q^{-274}-2q^{-276}+4q^{-278}+12q^{-280}+15q^{-282}+6q^{-284}-9q^{-286}-18q^{-288}-17q^{-290}-10q^{-292}+12q^{-294}+25q^{-296}+22q^{-298}+5q^{-300}-15q^{-302}-24q^{-304}-25q^{-306}-3q^{-308}+15q^{-310}+22q^{-312}+13q^{-314}-q^{-316}-12q^{-318}-18q^{-320}-7q^{-322}+7q^{-326}+7q^{-328}+4q^{-330}+3q^{-332}-3q^{-334}-3q^{-338}-4q^{-340}-4q^{-342}+6q^{-346}+3q^{-348}+6q^{-350}-4q^{-354}-6q^{-356}-3q^{-358}+2q^{-360}+2q^{-362}+6q^{-364}+3q^{-366}-3q^{-370}-3q^{-372}-q^{-374}-q^{-376}+3q^{-378}+2q^{-380}+2q^{-382}-q^{-388}-2q^{-390}-q^{-394}+q^{-396}-q^{-404}-q^{-408}+q^{-420}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-10}+q^{-14}+q^{-18}+q^{-20}+q^{-22}+2q^{-24}-q^{-30}-q^{-32}-q^{-34}-q^{-36}$
1,1	$q^{-20}+2q^{-24}-2q^{-26}+6q^{-28}-2q^{-30}+10q^{-32}-4q^{-34}+9q^{-36}-2q^{-38}+4q^{-40}-4q^{-44}+6q^{-46}-8q^{-48}+8q^{-50}-11q^{-52}+8q^{-54}-10q^{-56}+4q^{-58}-8q^{-60}-2q^{-64}-2q^{-66}+2q^{-68}+2q^{-72}+2q^{-74}+q^{-76}-2q^{-78}-2q^{-82}+3q^{-84}-4q^{-86}+2q^{-88}-2q^{-90}+2q^{-92}-2q^{-94}+2q^{-96}+q^{-100}$
2,0	$q^{-20}+q^{-26}+2q^{-28}+q^{-30}+q^{-32}+2q^{-34}+2q^{-36}+q^{-42}+q^{-44}+q^{-46}+2q^{-48}+q^{-50}-q^{-58}-3q^{-66}-3q^{-68}-3q^{-70}-3q^{-72}-3q^{-74}-2q^{-76}+q^{-80}+2q^{-82}+q^{-84}+2q^{-86}+q^{-88}+q^{-90}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-20}+q^{-24}+q^{-26}+q^{-30}+2q^{-32}+2q^{-34}+4q^{-36}+3q^{-38}+2q^{-40}+2q^{-42}+q^{-44}-2q^{-46}-q^{-48}-2q^{-50}-3q^{-52}-3q^{-54}-2q^{-56}-q^{-58}-q^{-60}+q^{-64}-q^{-66}-q^{-68}+q^{-70}-q^{-72}-q^{-74}+q^{-76}+q^{-78}+q^{-82}$
1,0,0	$q^{-15}+q^{-19}+q^{-23}+2q^{-27}+2q^{-29}+2q^{-31}+2q^{-33}-2q^{-39}-q^{-41}-2q^{-43}-q^{-45}-q^{-47}$
1,0,1	$q^{-30}+2q^{-34}+q^{-38}+3q^{-40}+2q^{-42}+9q^{-44}+5q^{-46}+8q^{-48}+7q^{-50}+5q^{-52}+7q^{-54}+2q^{-56}+5q^{-58}-q^{-60}+4q^{-62}-4q^{-64}-2q^{-66}-7q^{-68}-12q^{-70}-7q^{-72}-14q^{-74}-4q^{-76}-9q^{-78}+q^{-80}-3q^{-82}+2q^{-84}+3q^{-86}-2q^{-88}+6q^{-90}-2q^{-92}+6q^{-94}+q^{-96}+3q^{-98}+2q^{-100}-q^{-102}-2q^{-104}-q^{-106}+q^{-108}-2q^{-110}+3q^{-112}-2q^{-114}-q^{-116}+q^{-118}-2q^{-120}+q^{-122}-q^{-124}+2q^{-128}+q^{-132}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-30}+q^{-34}+q^{-36}+q^{-38}+2q^{-42}+2q^{-44}+3q^{-46}+4q^{-48}+4q^{-50}+5q^{-52}+5q^{-54}+4q^{-56}+3q^{-58}+3q^{-60}+2q^{-62}-q^{-64}-3q^{-66}-4q^{-68}-5q^{-70}-7q^{-72}-5q^{-74}-4q^{-76}-4q^{-78}-2q^{-80}-q^{-82}-2q^{-84}-2q^{-86}+q^{-94}+3q^{-96}+2q^{-98}+q^{-100}+q^{-102}+q^{-104}$
1,0,0,0	$q^{-20}+q^{-24}+q^{-28}+q^{-32}+2q^{-34}+2q^{-36}+3q^{-38}+2q^{-40}+2q^{-42}-2q^{-48}-2q^{-50}-2q^{-52}-2q^{-54}-q^{-56}-q^{-58}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-20}+q^{-24}-q^{-26}+2q^{-28}-q^{-30}+2q^{-32}+2q^{-36}+q^{-38}+2q^{-42}-q^{-44}+2q^{-46}-3q^{-48}+2q^{-50}-3q^{-52}+q^{-54}-2q^{-56}+q^{-58}-q^{-60}+q^{-64}-q^{-66}+q^{-68}-q^{-70}+q^{-72}-q^{-74}+q^{-76}-q^{-78}-q^{-82}$
1,0	$q^{-30}+q^{-38}+q^{-40}-q^{-44}+q^{-46}+2q^{-48}+q^{-50}-q^{-52}+q^{-54}+2q^{-56}+3q^{-58}+q^{-60}+2q^{-66}+q^{-68}-q^{-72}-q^{-78}-q^{-80}-q^{-82}-q^{-86}-2q^{-88}-2q^{-90}-q^{-96}-q^{-98}+q^{-102}-q^{-106}-q^{-108}+q^{-112}-q^{-116}-q^{-118}+q^{-122}+q^{-124}+q^{-132}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-30}+q^{-34}+2q^{-38}-q^{-40}+2q^{-42}+3q^{-46}+2q^{-48}+3q^{-50}+3q^{-52}+4q^{-54}+4q^{-56}+2q^{-58}+3q^{-60}-q^{-62}+q^{-64}-4q^{-66}-q^{-68}-5q^{-70}-2q^{-72}-5q^{-74}-q^{-76}-3q^{-78}-q^{-82}+q^{-90}-q^{-92}-q^{-96}+q^{-98}-q^{-100}-q^{-104}+q^{-106}+q^{-110}+q^{-114}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-50}+q^{-54}-q^{-56}+q^{-58}+3q^{-64}-2q^{-66}+3q^{-68}-q^{-70}+q^{-72}+2q^{-74}-3q^{-76}+3q^{-78}-q^{-80}+q^{-82}+q^{-84}-q^{-86}+2q^{-88}+q^{-90}+2q^{-94}-q^{-96}+q^{-98}+2q^{-100}-2q^{-102}+3q^{-104}-q^{-106}+2q^{-108}-q^{-112}+2q^{-114}-2q^{-116}+3q^{-118}-2q^{-120}+q^{-124}-2q^{-126}+q^{-128}-q^{-130}-q^{-132}+q^{-134}-q^{-136}-q^{-138}-q^{-142}-q^{-146}-2q^{-148}-q^{-152}-q^{-156}-q^{-160}-q^{-162}-q^{-164}-q^{-166}+2q^{-168}-2q^{-170}+q^{-172}+q^{-178}-q^{-180}+q^{-182}-q^{-184}+q^{-186}-q^{-190}+q^{-192}+q^{-196}$

.

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 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+9z^{4}+9z^{2}+1$

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]=

{ 19, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-7}+2z^{7}a^{-9}+z^{7}a^{-11}+z^{6}a^{-6}-5z^{6}a^{-8}-5z^{6}a^{-10}+z^{6}a^{-12}-4z^{5}a^{-7}-8z^{5}a^{-9}-3z^{5}a^{-11}+z^{5}a^{-13}-5z^{4}a^{-6}+9z^{4}a^{-8}+11z^{4}a^{-10}-2z^{4}a^{-12}+z^{4}a^{-14}+3z^{3}a^{-7}+9z^{3}a^{-9}+4z^{3}a^{-11}-z^{3}a^{-13}+z^{3}a^{-15}+6z^{2}a^{-6}-9z^{2}a^{-8}-11z^{2}a^{-10}+3z^{2}a^{-12}-z^{2}a^{-14}-4za^{-9}-za^{-11}+za^{-13}-2za^{-15}-a^{-6}+3a^{-8}+3a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(9, 26)

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 $36$ $208$ $648$ $1578$ $246$ $7488$ ${\frac {39616}{3}}$ ${\frac {7072}{3}}$ $1744$ $7776$ $21632$ $56808$ $8856$ ${\frac {1125053}{10}}$ ${\frac {16214}{5}}$ ${\frac {218942}{5}}$ ${\frac {1313}{2}}$ ${\frac {57373}{10}}$

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=$6 is the signature of 9 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 25                   1 -1 23                     0 21               2 1   -1 19             1       1 17           2 2       0 15         1 1         0 13       1 2           1 11     1 1             0 9     1               1 7 1 1                 0 5 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=5$ $i=7$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{33}-q^{32}+2q^{30}-2q^{29}-q^{28}+3q^{27}-4q^{26}+5q^{24}-6q^{23}+q^{22}+5q^{21}-6q^{20}+6q^{18}-5q^{17}-q^{16}+6q^{15}-4q^{14}-2q^{13}+5q^{12}-2q^{11}-2q^{10}+3q^{9}-q^{7}+q^{6}$
3	$-q^{63}+q^{62}-2q^{59}+2q^{58}+q^{57}+q^{56}-4q^{55}+q^{54}+3q^{53}+2q^{52}-5q^{51}-q^{50}+3q^{49}+3q^{48}-3q^{47}-4q^{46}+3q^{45}+2q^{44}-4q^{42}+q^{41}+2q^{40}-3q^{38}+2q^{37}+q^{36}-2q^{35}-2q^{34}+4q^{33}-q^{32}-3q^{31}+6q^{29}-3q^{28}-4q^{27}+q^{26}+7q^{25}-3q^{24}-5q^{23}+7q^{21}-q^{20}-4q^{19}-2q^{18}+5q^{17}+q^{16}-2q^{15}-2q^{14}+2q^{13}+q^{12}-q^{10}+q^{9}$
4	$q^{102}-q^{101}+2q^{97}-3q^{96}+5q^{92}-5q^{91}-q^{90}-2q^{89}+q^{88}+10q^{87}-6q^{86}-2q^{85}-7q^{84}+2q^{83}+17q^{82}-5q^{81}-4q^{80}-14q^{79}-q^{78}+26q^{77}-q^{76}-4q^{75}-23q^{74}-5q^{73}+32q^{72}+3q^{71}-q^{70}-27q^{69}-9q^{68}+31q^{67}+5q^{66}+q^{65}-27q^{64}-9q^{63}+30q^{62}+3q^{61}+q^{60}-24q^{59}-9q^{58}+29q^{57}+q^{56}+q^{55}-20q^{54}-9q^{53}+26q^{52}-2q^{51}+2q^{50}-14q^{49}-8q^{48}+21q^{47}-5q^{46}+2q^{45}-8q^{44}-6q^{43}+17q^{42}-8q^{41}+q^{40}-4q^{39}-3q^{38}+15q^{37}-8q^{36}-q^{35}-4q^{34}-2q^{33}+14q^{32}-5q^{31}-q^{30}-5q^{29}-4q^{28}+11q^{27}-q^{26}+q^{25}-4q^{24}-5q^{23}+6q^{22}+2q^{20}-q^{19}-3q^{18}+2q^{17}+q^{15}-q^{13}+q^{12}$
5	$-q^{150}+q^{149}-q^{144}+2q^{143}-q^{141}-2q^{138}+4q^{137}+q^{136}-2q^{135}-2q^{133}-4q^{132}+5q^{131}+4q^{130}-q^{129}-5q^{127}-6q^{126}+4q^{125}+9q^{124}+2q^{123}-q^{122}-11q^{121}-9q^{120}+6q^{119}+16q^{118}+7q^{117}-4q^{116}-22q^{115}-14q^{114}+9q^{113}+29q^{112}+17q^{111}-10q^{110}-34q^{109}-24q^{108}+10q^{107}+40q^{106}+31q^{105}-12q^{104}-42q^{103}-31q^{102}+6q^{101}+43q^{100}+38q^{99}-8q^{98}-44q^{97}-33q^{96}+4q^{95}+41q^{94}+38q^{93}-7q^{92}-42q^{91}-32q^{90}+4q^{89}+39q^{88}+35q^{87}-6q^{86}-37q^{85}-32q^{84}+2q^{83}+35q^{82}+34q^{81}-2q^{80}-32q^{79}-31q^{78}-4q^{77}+28q^{76}+34q^{75}+2q^{74}-24q^{73}-28q^{72}-10q^{71}+20q^{70}+31q^{69}+7q^{68}-16q^{67}-22q^{66}-13q^{65}+10q^{64}+24q^{63}+8q^{62}-8q^{61}-14q^{60}-11q^{59}+4q^{58}+15q^{57}+4q^{56}-3q^{55}-7q^{54}-7q^{53}+3q^{52}+10q^{51}-3q^{49}-5q^{48}-4q^{47}+3q^{46}+10q^{45}+q^{44}-3q^{43}-6q^{42}-5q^{41}+9q^{39}+4q^{38}+q^{37}-4q^{36}-7q^{35}-3q^{34}+5q^{33}+3q^{32}+4q^{31}-4q^{29}-4q^{28}+2q^{27}+2q^{25}+2q^{24}-q^{23}-2q^{22}+q^{21}+q^{18}-q^{16}+q^{15}$
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{207}-q^{206}-q^{201}+2 q^{200}-2 q^{199}+q^{198}+q^{195}-3 q^{194}+3 q^{193}-4 q^{192}+2 q^{191}+q^{190}+3 q^{188}-3 q^{187}+4 q^{186}-8 q^{185}+2 q^{184}-q^{183}+6 q^{181}+7 q^{179}-12 q^{178}+2 q^{177}-6 q^{176}-3 q^{175}+8 q^{174}+4 q^{173}+13 q^{172}-15 q^{171}+5 q^{170}-12 q^{169}-7 q^{168}+8 q^{167}+5 q^{166}+16 q^{165}-18 q^{164}+11 q^{163}-11 q^{162}-4 q^{161}+8 q^{160}-2 q^{159}+9 q^{158}-29 q^{157}+17 q^{156}+10 q^{154}+17 q^{153}-9 q^{152}-9 q^{151}-51 q^{150}+18 q^{149}+9 q^{148}+30 q^{147}+34 q^{146}-6 q^{145}-22 q^{144}-73 q^{143}+11 q^{142}+8 q^{141}+39 q^{140}+49 q^{139}+3 q^{138}-25 q^{137}-81 q^{136}+5 q^{135}+3 q^{134}+38 q^{133}+53 q^{132}+8 q^{131}-24 q^{130}-79 q^{129}+5 q^{128}+2 q^{127}+35 q^{126}+50 q^{125}+8 q^{124}-21 q^{123}-76 q^{122}+5 q^{121}+q^{120}+33 q^{119}+46 q^{118}+8 q^{117}-13 q^{116}-72 q^{115}+q^{114}-5 q^{113}+27 q^{112}+44 q^{111}+14 q^{110}+q^{109}-66 q^{108}-8 q^{107}-15 q^{106}+18 q^{105}+42 q^{104}+23 q^{103}+17 q^{102}-57 q^{101}-18 q^{100}-28 q^{99}+6 q^{98}+38 q^{97}+32 q^{96}+34 q^{95}-44 q^{94}-22 q^{93}-39 q^{92}-8 q^{91}+28 q^{90}+33 q^{89}+47 q^{88}-27 q^{87}-17 q^{86}-40 q^{85}-19 q^{84}+12 q^{83}+24 q^{82}+49 q^{81}-12 q^{80}-5 q^{79}-30 q^{78}-20 q^{77}-2 q^{76}+9 q^{75}+40 q^{74}-7 q^{73}+5 q^{72}-16 q^{71}-12 q^{70}-7 q^{69}-q^{68}+29 q^{67}-9 q^{66}+5 q^{65}-8 q^{64}-4 q^{63}-6 q^{62}-2 q^{61}+24 q^{60}-9 q^{59}+3 q^{58}-7 q^{57}-3 q^{56}-8 q^{55}-2 q^{54}+22 q^{53}-4 q^{52}+5 q^{51}-4 q^{50}-3 q^{49}-12 q^{48}-6 q^{47}+15 q^{46}-q^{45}+8 q^{44}+q^{43}+q^{42}-10 q^{41}-8 q^{40}+7 q^{39}-3 q^{38}+5 q^{37}+3 q^{36}+4 q^{35}-4 q^{34}-5 q^{33}+3 q^{32}-3 q^{31}+q^{30}+q^{29}+3 q^{28}-q^{27}-2 q^{26}+2 q^{25}-q^{24}+q^{21}-q^{19}+q^{18}}
7	Not Available

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 3]]
Out[2]=	PD[X[8, 2, 9, 1], X[12, 4, 13, 3], X[18, 10, 1, 9], X[10, 18, 11, 17], X[14, 6, 15, 5], X[16, 8, 17, 7], X[2, 12, 3, 11], X[4, 14, 5, 13], X[6, 16, 7, 15]]
In[3]:=	GaussCode[Knot[9, 3]]
Out[3]=	GaussCode[1, -7, 2, -8, 5, -9, 6, -1, 3, -4, 7, -2, 8, -5, 9, -6, 4, -3]
In[4]:=	DTCode[Knot[9, 3]]
Out[4]=	DTCode[8, 12, 14, 16, 18, 2, 4, 6, 10]
In[5]:=	br = BR[Knot[9, 3]]
Out[5]=	BR[3, {1, 1, 1, 1, 1, 1, 1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[9, 3]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[9, 3]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 3]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 3]][t]
Out[10]=	2 3 3 2 3 -3 + -- - -- + - + 3 t - 3 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[9, 3]][z]
Out[11]=	2 4 6 1 + 9 z + 9 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 3]}
In[13]:=	{KnotDet[Knot[9, 3]], KnotSignature[Knot[9, 3]]}
Out[13]=	{19, 6}
In[14]:=	Jones[Knot[9, 3]][q]
Out[14]=	3 4 5 6 7 8 9 10 11 12 q - q + 2 q - 2 q + 3 q - 3 q + 3 q - 2 q + q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 3]}
In[16]:=	A2Invariant[Knot[9, 3]][q]
Out[16]=	10 14 18 20 22 24 30 32 34 36 q + q + q + q + q + 2 q - q - q - q - q
In[17]:=	HOMFLYPT[Knot[9, 3]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 -3 3 -6 4 z 7 z 6 z z 5 z 5 z z z --- + -- + a - ---- + ---- + ---- - --- + ---- + ---- + -- + -- 10 8 10 8 6 10 8 6 8 6 a a a a a a a a a a
In[18]:=	Kauffman[Knot[9, 3]][a, z]
Out[18]=	2 2 2 2 3 3 -6 2 z z z 4 z z 3 z 11 z 9 z --- + -- - a - --- + --- - --- - --- - --- + ---- - ----- - ---- + 10 8 15 13 11 9 14 12 10 8 a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 6 z z z 4 z 9 z 3 z z 2 z 11 z 9 z ---- + --- - --- + ---- + ---- + ---- + --- - ---- + ----- + ---- - 6 15 13 11 9 7 14 12 10 8 a a a a a a a a a a 4 5 5 5 5 6 6 6 6 7 5 z z 3 z 8 z 4 z z 5 z 5 z z z ---- + --- - ---- - ---- - ---- + --- - ---- - ---- + -- + --- + 6 13 11 9 7 12 10 8 6 11 a a a a a a a a a a 7 7 8 8 2 z z z z ---- + -- + --- + -- 9 7 10 8 a a a a
In[19]:=	{Vassiliev[2][Knot[9, 3]], Vassiliev[3][Knot[9, 3]]}
Out[19]=	{9, 26}
In[20]:=	Kh[Knot[9, 3]][q, t]
Out[20]=	5 7 7 9 2 11 2 11 3 13 3 13 4 15 4 q + q + q t + q t + q t + q t + q t + 2 q t + q t + 15 5 17 5 17 6 19 6 21 7 21 8 25 9 q t + 2 q t + 2 q t + q t + 2 q t + q t + q t
In[21]:=	ColouredJones[Knot[9, 3], 2][q]
Out[21]=	6 7 9 10 11 12 13 14 15 16 q - q + 3 q - 2 q - 2 q + 5 q - 2 q - 4 q + 6 q - q - 17 18 20 21 22 23 24 26 27 5 q + 6 q - 6 q + 5 q + q - 6 q + 5 q - 4 q + 3 q - 28 29 30 32 33 q - 2 q + 2 q - q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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