10 139

10_138

10_140

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10 139 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X11,19,12,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8 X13,1,14,20 X19,13,20,12 X2,10,3,9
Gauss code	1, -10, 2, -1, -4, 6, -5, 7, 10, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8
Dowker-Thistlethwaite code	4 10 -14 -16 2 -18 -20 -6 -8 -12
Conway Notation	[4,3,3-]

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 4 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [7][-16] Hyperbolic Volume 4.85117 A-Polynomial See Data:10 139/A-polynomial

[edit Notes for 10 139's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $4$ Topological 4 genus $[3,4]$ Concordance genus $4$ Rasmussen s-Invariant -8

[edit Notes for 10 139's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-t^{3}+2t-3+2t^{-1}-t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+7z^{6}+14z^{4}+9z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 3, 6 }
Jones polynomial	$-q^{12}+q^{11}-q^{10}+q^{9}-q^{8}+q^{6}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-13z^{2}a^{-10}+z^{2}a^{-12}+6a^{-8}-6a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+20z^{4}a^{-10}+z^{4}a^{-14}+13z^{3}a^{-9}+13z^{3}a^{-11}+z^{3}a^{-13}+z^{3}a^{-15}-21z^{2}a^{-8}-19z^{2}a^{-10}-2z^{2}a^{-14}-6za^{-9}-5za^{-11}-za^{-13}-2za^{-15}+6a^{-8}+6a^{-10}+a^{-12}$
The A2 invariant	$q^{-14}+q^{-16}+2q^{-18}+2q^{-20}+q^{-22}-q^{-28}-q^{-32}-q^{-34}-q^{-36}-q^{-38}+q^{-40}$
The G2 invariant	$q^{-70}+q^{-72}+q^{-74}+q^{-76}+2q^{-80}+3q^{-82}+q^{-84}+q^{-86}+q^{-88}+3q^{-90}+3q^{-92}+2q^{-94}-2q^{-96}+2q^{-98}+3q^{-100}+q^{-102}-3q^{-106}+q^{-108}+2q^{-110}-2q^{-112}-3q^{-114}-3q^{-116}-q^{-118}+4q^{-120}-3q^{-122}-3q^{-124}-q^{-126}-q^{-128}+2q^{-130}-3q^{-132}-q^{-134}+2q^{-140}-q^{-152}-q^{-156}-q^{-158}+2q^{-160}-2q^{-162}-q^{-164}-3q^{-168}+q^{-170}-2q^{-172}+q^{-176}-q^{-178}+2q^{-180}+2q^{-186}-q^{-188}-q^{-190}+q^{-192}+q^{-196}$

Further Quantum Invariants

Further quantum knot invariants for 10_139.

A1 Invariants.

Weight	Invariant
1	$q^{-7}+q^{-9}+q^{-11}+q^{-13}-q^{-15}-q^{-25}$
2	$q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}+q^{-26}-q^{-28}-q^{-30}-q^{-36}+2q^{-40}-q^{-44}-q^{-48}-q^{-50}-q^{-52}-q^{-56}+2q^{-60}-q^{-64}+q^{-68}$
3	$q^{-21}+q^{-23}+q^{-25}+q^{-27}+q^{-29}+q^{-31}+q^{-33}+q^{-35}+q^{-37}+q^{-39}-q^{-41}-q^{-43}-q^{-45}-q^{-53}-q^{-55}+2q^{-59}+2q^{-61}-2q^{-65}-q^{-67}+2q^{-69}-2q^{-73}-4q^{-75}+q^{-79}-q^{-81}-q^{-83}+q^{-87}-q^{-101}+3q^{-105}+q^{-107}-2q^{-109}+3q^{-113}+2q^{-115}-2q^{-117}-3q^{-119}+q^{-123}+q^{-125}-q^{-129}$
5	$q^{-35}+q^{-37}+q^{-39}+q^{-41}+q^{-43}+q^{-45}+q^{-47}+q^{-49}+q^{-51}+q^{-53}+q^{-55}+q^{-57}+q^{-59}+q^{-61}+q^{-63}+q^{-65}-q^{-67}-q^{-69}-q^{-71}-q^{-73}-q^{-75}-q^{-87}-q^{-89}-q^{-91}-q^{-93}+2q^{-97}+2q^{-99}+2q^{-101}+2q^{-103}-2q^{-107}-2q^{-109}-2q^{-111}-q^{-113}+2q^{-115}+2q^{-117}+2q^{-119}-2q^{-123}-5q^{-125}-5q^{-127}-4q^{-129}+3q^{-133}+5q^{-135}+3q^{-137}-2q^{-139}-6q^{-141}-7q^{-143}-3q^{-145}+2q^{-147}+7q^{-149}+9q^{-151}+4q^{-153}-4q^{-155}-9q^{-157}-7q^{-159}+7q^{-163}+11q^{-165}+4q^{-167}-6q^{-169}-11q^{-171}-7q^{-173}-q^{-175}+9q^{-177}+10q^{-179}-6q^{-183}-7q^{-185}+q^{-187}+7q^{-189}+7q^{-191}+2q^{-193}-4q^{-195}-4q^{-197}+q^{-199}+3q^{-201}+2q^{-203}-q^{-207}-q^{-209}+q^{-229}+q^{-231}-q^{-233}-3q^{-235}-2q^{-237}+q^{-239}+4q^{-241}+2q^{-243}-3q^{-245}-9q^{-247}-8q^{-249}+q^{-251}+10q^{-253}+12q^{-255}+2q^{-257}-9q^{-259}-12q^{-261}-7q^{-263}+6q^{-265}+13q^{-267}+10q^{-269}+q^{-271}-7q^{-273}-9q^{-275}-7q^{-277}+6q^{-281}+7q^{-283}+3q^{-285}-3q^{-289}-4q^{-291}-q^{-293}+q^{-297}+q^{-299}+q^{-301}-q^{-305}$
6	$q^{-42}+q^{-44}+q^{-46}+q^{-48}+q^{-50}+q^{-52}+q^{-54}+q^{-56}+q^{-58}+q^{-60}+q^{-62}+q^{-64}+q^{-66}+q^{-68}+q^{-70}+q^{-72}+q^{-74}+q^{-76}+q^{-78}-q^{-80}-q^{-82}-q^{-84}-q^{-86}-q^{-88}-q^{-90}-q^{-104}-q^{-106}-q^{-108}-q^{-110}-q^{-112}+2q^{-116}+2q^{-118}+2q^{-120}+2q^{-122}+2q^{-124}-2q^{-128}-2q^{-130}-2q^{-132}-2q^{-134}-q^{-136}+2q^{-138}+2q^{-140}+2q^{-142}+2q^{-144}-2q^{-148}-5q^{-150}-5q^{-152}-5q^{-154}-4q^{-156}+3q^{-160}+5q^{-162}+5q^{-164}+3q^{-166}-2q^{-168}-6q^{-170}-8q^{-172}-7q^{-174}-3q^{-176}+2q^{-178}+7q^{-180}+11q^{-182}+9q^{-184}+4q^{-186}-4q^{-188}-10q^{-190}-12q^{-192}-9q^{-194}+9q^{-198}+15q^{-200}+14q^{-202}+6q^{-204}-7q^{-206}-16q^{-208}-18q^{-210}-13q^{-212}+q^{-214}+15q^{-216}+22q^{-218}+15q^{-220}-13q^{-224}-22q^{-226}-17q^{-228}+19q^{-232}+25q^{-234}+17q^{-236}+q^{-238}-14q^{-240}-23q^{-242}-14q^{-244}+6q^{-246}+17q^{-248}+17q^{-250}+7q^{-252}-8q^{-254}-18q^{-256}-13q^{-258}+q^{-260}+11q^{-262}+11q^{-264}+6q^{-266}-3q^{-268}-10q^{-270}-6q^{-272}+q^{-274}+5q^{-276}+4q^{-278}+q^{-280}-q^{-282}-2q^{-284}-q^{-286}-q^{-294}-q^{-296}-q^{-298}-q^{-300}-q^{-302}-q^{-304}-q^{-306}-q^{-314}-2q^{-316}-q^{-318}+q^{-320}+4q^{-322}+5q^{-324}+q^{-326}-5q^{-328}-11q^{-330}-6q^{-332}+4q^{-334}+13q^{-336}+17q^{-338}+6q^{-340}-10q^{-342}-23q^{-344}-19q^{-346}-2q^{-348}+18q^{-350}+29q^{-352}+23q^{-354}+q^{-356}-22q^{-358}-28q^{-360}-21q^{-362}-3q^{-364}+18q^{-366}+27q^{-368}+21q^{-370}+2q^{-372}-12q^{-374}-19q^{-376}-19q^{-378}-8q^{-380}+5q^{-382}+12q^{-384}+12q^{-386}+9q^{-388}+2q^{-390}-5q^{-392}-8q^{-394}-6q^{-396}-3q^{-398}+3q^{-402}+4q^{-404}+2q^{-406}-q^{-412}-q^{-414}-q^{-416}+q^{-420}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-14}+q^{-16}+2q^{-18}+2q^{-20}+q^{-22}-q^{-28}-q^{-32}-q^{-34}-q^{-36}-q^{-38}+q^{-40}$
1,1	$q^{-28}+2q^{-30}+4q^{-32}+6q^{-34}+5q^{-36}+6q^{-38}+4q^{-40}+4q^{-42}+q^{-44}-6q^{-48}-4q^{-50}-9q^{-52}-4q^{-54}-6q^{-56}-4q^{-58}+3q^{-60}-2q^{-62}+6q^{-64}-4q^{-66}+6q^{-68}-2q^{-70}+2q^{-72}-2q^{-74}-2q^{-80}+4q^{-82}-4q^{-84}+4q^{-86}-2q^{-88}+2q^{-90}-2q^{-92}-2q^{-94}+2q^{-96}+q^{-100}$
2,0	$q^{-28}+q^{-30}+2q^{-32}+2q^{-34}+3q^{-36}+3q^{-38}+4q^{-40}+2q^{-42}+q^{-44}-2q^{-50}-2q^{-52}-q^{-54}+q^{-58}-q^{-64}-3q^{-66}-4q^{-68}-6q^{-70}-4q^{-72}-2q^{-74}+2q^{-78}+3q^{-80}+3q^{-82}+2q^{-84}+2q^{-86}-q^{-88}+q^{-90}-q^{-96}-q^{-98}+q^{-100}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-28}+q^{-30}+3q^{-32}+4q^{-34}+5q^{-36}+4q^{-38}+5q^{-40}+q^{-42}+q^{-44}-2q^{-46}-4q^{-48}-4q^{-50}-6q^{-52}-4q^{-54}-3q^{-56}-q^{-58}+q^{-60}+2q^{-62}+q^{-64}+q^{-66}-q^{-70}-q^{-74}+q^{-78}+q^{-82}$
1,0,0	$q^{-21}+q^{-23}+2q^{-25}+3q^{-27}+2q^{-29}+2q^{-31}+q^{-33}-q^{-37}-q^{-39}-2q^{-41}-2q^{-43}-2q^{-45}-q^{-47}+q^{-53}$
1,0,1	$q^{-42}+2q^{-44}+5q^{-46}+9q^{-48}+12q^{-50}+15q^{-52}+17q^{-54}+14q^{-56}+14q^{-58}+7q^{-60}+2q^{-62}-7q^{-64}-15q^{-66}-19q^{-68}-26q^{-70}-22q^{-72}-21q^{-74}-12q^{-76}-7q^{-78}+2q^{-80}+8q^{-82}+8q^{-84}+14q^{-86}+7q^{-88}+8q^{-90}+5q^{-92}+3q^{-94}-4q^{-98}-q^{-100}-6q^{-102}-2q^{-104}-q^{-106}+q^{-108}+2q^{-110}+q^{-112}+4q^{-114}-2q^{-116}-q^{-120}-2q^{-122}-q^{-124}+2q^{-128}+q^{-132}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-42}+q^{-44}+3q^{-46}+5q^{-48}+7q^{-50}+8q^{-52}+11q^{-54}+9q^{-56}+9q^{-58}+5q^{-60}+q^{-62}-4q^{-64}-8q^{-66}-13q^{-68}-14q^{-70}-12q^{-72}-10q^{-74}-6q^{-76}-2q^{-78}+4q^{-80}+4q^{-82}+4q^{-84}+4q^{-86}+q^{-88}-2q^{-90}-q^{-92}-q^{-94}-q^{-96}+q^{-98}+3q^{-100}+2q^{-102}+q^{-104}+q^{-106}+q^{-108}-q^{-110}-q^{-112}$
1,0,0,0	$q^{-28}+q^{-30}+2q^{-32}+3q^{-34}+3q^{-36}+3q^{-38}+3q^{-40}+q^{-42}+q^{-44}-q^{-46}-2q^{-48}-3q^{-50}-3q^{-52}-3q^{-54}-2q^{-56}-q^{-58}+q^{-62}+q^{-66}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-28}+q^{-30}+q^{-32}+2q^{-34}+q^{-36}+2q^{-38}+q^{-40}+q^{-42}+q^{-44}-2q^{-50}-2q^{-54}-q^{-56}-q^{-58}-q^{-60}-q^{-64}+q^{-66}+q^{-70}+q^{-74}-q^{-78}-q^{-82}$
1,0	$q^{-42}+q^{-46}+q^{-48}+2q^{-50}+q^{-52}+3q^{-54}+2q^{-56}+2q^{-58}+q^{-60}+2q^{-62}+q^{-64}+q^{-66}-q^{-68}-q^{-74}-2q^{-76}-q^{-78}-2q^{-82}-2q^{-84}-2q^{-86}-q^{-88}-2q^{-90}-q^{-92}-q^{-94}+q^{-100}+q^{-106}+q^{-108}-q^{-112}-q^{-120}+q^{-124}+q^{-132}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-42}+q^{-44}+2q^{-46}+4q^{-48}+4q^{-50}+6q^{-52}+6q^{-54}+6q^{-56}+5q^{-58}+4q^{-60}+q^{-62}-2q^{-64}-4q^{-66}-7q^{-68}-6q^{-70}-8q^{-72}-6q^{-74}-5q^{-76}-2q^{-78}+q^{-82}+2q^{-84}+2q^{-86}+2q^{-88}+q^{-90}+q^{-92}-q^{-94}-q^{-98}-q^{-102}+q^{-110}+q^{-114}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-70}+q^{-72}+q^{-74}+q^{-76}+2q^{-80}+3q^{-82}+q^{-84}+q^{-86}+q^{-88}+3q^{-90}+3q^{-92}+2q^{-94}-2q^{-96}+2q^{-98}+3q^{-100}+q^{-102}-3q^{-106}+q^{-108}+2q^{-110}-2q^{-112}-3q^{-114}-3q^{-116}-q^{-118}+4q^{-120}-3q^{-122}-3q^{-124}-q^{-126}-q^{-128}+2q^{-130}-3q^{-132}-q^{-134}+2q^{-140}-q^{-152}-q^{-156}-q^{-158}+2q^{-160}-2q^{-162}-q^{-164}-3q^{-168}+q^{-170}-2q^{-172}+q^{-176}-q^{-178}+2q^{-180}+2q^{-186}-q^{-188}-q^{-190}+q^{-192}+q^{-196}$

.

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["10 139"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+7z^{6}+14z^{4}+9z^{2}+1$

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]=

{ 3, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{12}+q^{11}-q^{10}+q^{9}-q^{8}+q^{6}+q^{4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-13z^{2}a^{-10}+z^{2}a^{-12}+6a^{-8}-6a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+20z^{4}a^{-10}+z^{4}a^{-14}+13z^{3}a^{-9}+13z^{3}a^{-11}+z^{3}a^{-13}+z^{3}a^{-15}-21z^{2}a^{-8}-19z^{2}a^{-10}-2z^{2}a^{-14}-6za^{-9}-5za^{-11}-za^{-13}-2za^{-15}+6a^{-8}+6a^{-10}+a^{-12}$

"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, 25)

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 $36$ $200$ $648$ $1466$ $206$ $7200$ ${\frac {35888}{3}}$ ${\frac {6272}{3}}$ $1384$ $7776$ $20000$ $52776$ $7416$ ${\frac {1001773}{10}}$ ${\frac {77402}{15}}$ ${\frac {523826}{15}}$ ${\frac {3059}{6}}$ ${\frac {43373}{10}}$

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 10 139. 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               1 1   0 19           1 1       0 17           1 1       0 15       1 1 1         -1 13         1           1 11     1               1 9 1                   1 7 1                   1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{33}-q^{32}-q^{31}+2q^{30}+q^{29}-3q^{28}+q^{27}+2q^{26}-4q^{25}+q^{24}+2q^{23}-3q^{22}+3q^{20}-q^{19}-2q^{18}+2q^{17}-2q^{15}+q^{14}+q^{11}+q^{8}$
3	$-q^{63}+q^{62}+q^{61}-2q^{59}-2q^{58}+2q^{57}+4q^{56}-q^{55}-5q^{54}+7q^{52}+q^{51}-8q^{50}-q^{49}+8q^{48}+q^{47}-8q^{46}-q^{45}+8q^{44}+q^{43}-7q^{42}-2q^{41}+7q^{40}+q^{39}-5q^{38}-3q^{37}+3q^{36}+3q^{35}-3q^{34}-q^{33}+2q^{31}-q^{30}+q^{29}-2q^{26}+q^{25}+q^{24}-2q^{22}+q^{20}+q^{16}+q^{12}$
4	$q^{102}-q^{101}-q^{100}+3q^{97}+q^{96}-q^{95}-3q^{94}-5q^{93}+3q^{92}+6q^{91}+3q^{90}-2q^{89}-9q^{88}-4q^{87}+7q^{86}+9q^{85}+q^{84}-11q^{83}-9q^{82}+8q^{81}+11q^{80}+2q^{79}-11q^{78}-10q^{77}+8q^{76}+12q^{75}+2q^{74}-11q^{73}-10q^{72}+8q^{71}+11q^{70}+2q^{69}-10q^{68}-10q^{67}+7q^{66}+9q^{65}+q^{64}-5q^{63}-9q^{62}+4q^{61}+5q^{60}+2q^{59}+2q^{58}-8q^{57}+q^{56}+5q^{53}-3q^{52}+q^{51}-3q^{50}-4q^{49}+4q^{48}+3q^{46}-q^{45}-4q^{44}+q^{43}-q^{42}+3q^{41}+q^{40}-2q^{39}+q^{38}-q^{37}+q^{36}-2q^{34}+q^{33}+q^{31}-2q^{29}+q^{26}+q^{21}+q^{16}$
5	$-q^{150}+q^{149}+q^{148}-q^{145}-2q^{144}-2q^{143}+2q^{142}+3q^{141}+3q^{140}+3q^{139}-3q^{138}-8q^{137}-5q^{136}+q^{135}+5q^{134}+11q^{133}+6q^{132}-5q^{131}-12q^{130}-12q^{129}+14q^{127}+17q^{126}+5q^{125}-14q^{124}-21q^{123}-9q^{122}+13q^{121}+23q^{120}+10q^{119}-12q^{118}-24q^{117}-12q^{116}+12q^{115}+25q^{114}+12q^{113}-12q^{112}-25q^{111}-12q^{110}+12q^{109}+25q^{108}+12q^{107}-12q^{106}-25q^{105}-12q^{104}+12q^{103}+24q^{102}+12q^{101}-11q^{100}-23q^{99}-11q^{98}+10q^{97}+19q^{96}+12q^{95}-5q^{94}-18q^{93}-11q^{92}+4q^{91}+11q^{90}+13q^{89}+q^{88}-8q^{87}-12q^{86}-6q^{85}+5q^{84}+9q^{83}+6q^{82}+2q^{81}-5q^{80}-10q^{79}-2q^{78}+2q^{77}+4q^{76}+7q^{75}+3q^{74}-5q^{73}-4q^{72}-3q^{71}-q^{70}+3q^{69}+4q^{68}-q^{67}+q^{66}-q^{65}-3q^{64}-2q^{61}+q^{60}+2q^{59}-q^{58}+2q^{57}-2q^{55}-2q^{54}+q^{53}-q^{52}+2q^{51}+2q^{50}-2q^{48}+q^{47}-q^{46}+q^{44}-2q^{42}+q^{41}+q^{38}-2q^{36}+q^{32}+q^{26}+q^{20}$
6	$q^{207}-q^{206}-q^{205}+q^{202}+3q^{200}+q^{199}-2q^{198}-3q^{197}-3q^{196}-2q^{195}-2q^{194}+6q^{193}+8q^{192}+5q^{191}-3q^{189}-9q^{188}-15q^{187}-5q^{186}+8q^{185}+12q^{184}+14q^{183}+16q^{182}-3q^{181}-24q^{180}-26q^{179}-10q^{178}+5q^{177}+20q^{176}+39q^{175}+19q^{174}-18q^{173}-37q^{172}-30q^{171}-12q^{170}+16q^{169}+52q^{168}+35q^{167}-7q^{166}-41q^{165}-39q^{164}-22q^{163}+11q^{162}+58q^{161}+41q^{160}-3q^{159}-42q^{158}-42q^{157}-24q^{156}+10q^{155}+59q^{154}+42q^{153}-3q^{152}-42q^{151}-43q^{150}-24q^{149}+10q^{148}+59q^{147}+42q^{146}-3q^{145}-42q^{144}-42q^{143}-24q^{142}+10q^{141}+58q^{140}+41q^{139}-2q^{138}-40q^{137}-39q^{136}-23q^{135}+6q^{134}+51q^{133}+37q^{132}+5q^{131}-31q^{130}-34q^{129}-23q^{128}-4q^{127}+37q^{126}+32q^{125}+15q^{124}-16q^{123}-24q^{122}-23q^{121}-15q^{120}+17q^{119}+23q^{118}+24q^{117}-q^{116}-8q^{115}-15q^{114}-21q^{113}-2q^{112}+6q^{111}+19q^{110}+8q^{109}+5q^{108}-14q^{106}-9q^{105}-8q^{104}+5q^{103}+3q^{102}+7q^{101}+9q^{100}-q^{99}-q^{98}-7q^{97}-q^{96}-6q^{95}-2q^{94}+6q^{93}+q^{92}+5q^{91}+q^{90}+4q^{89}-4q^{88}-6q^{87}+q^{86}-4q^{85}+q^{84}+6q^{82}-q^{80}+3q^{79}-4q^{78}-q^{77}-3q^{76}+2q^{75}-q^{74}-q^{73}+3q^{72}-q^{71}+q^{70}-q^{69}+2q^{68}-q^{67}-q^{66}-2q^{64}+q^{63}-q^{62}+2q^{61}+q^{60}+q^{59}-2q^{57}+q^{56}-q^{55}+q^{52}-2q^{50}+q^{49}+q^{45}-2q^{43}+q^{38}+q^{31}+q^{24}$
7	$-q^{273}+q^{272}+q^{271}-q^{268}-q^{266}-2q^{265}-q^{264}+2q^{263}+3q^{262}+4q^{261}+q^{260}+q^{258}-7q^{257}-9q^{256}-5q^{255}+5q^{253}+8q^{252}+10q^{251}+15q^{250}+4q^{249}-10q^{248}-16q^{247}-21q^{246}-15q^{245}-7q^{244}+10q^{243}+32q^{242}+35q^{241}+18q^{240}+2q^{239}-24q^{238}-46q^{237}-42q^{236}-25q^{235}+17q^{234}+50q^{233}+56q^{232}+48q^{231}+3q^{230}-47q^{229}-68q^{228}-68q^{227}-23q^{226}+41q^{225}+71q^{224}+84q^{223}+39q^{222}-33q^{221}-74q^{220}-94q^{219}-49q^{218}+28q^{217}+74q^{216}+100q^{215}+57q^{214}-26q^{213}-75q^{212}-102q^{211}-59q^{210}+25q^{209}+74q^{208}+104q^{207}+61q^{206}-25q^{205}-74q^{204}-105q^{203}-61q^{202}+25q^{201}+74q^{200}+105q^{199}+61q^{198}-25q^{197}-74q^{196}-105q^{195}-61q^{194}+25q^{193}+74q^{192}+105q^{191}+61q^{190}-25q^{189}-74q^{188}-104q^{187}-61q^{186}+25q^{185}+73q^{184}+102q^{183}+60q^{182}-23q^{181}-71q^{180}-99q^{179}-60q^{178}+20q^{177}+64q^{176}+94q^{175}+62q^{174}-15q^{173}-57q^{172}-88q^{171}-59q^{170}+5q^{169}+42q^{168}+80q^{167}+63q^{166}+4q^{165}-31q^{164}-66q^{163}-60q^{162}-16q^{161}+14q^{160}+51q^{159}+58q^{158}+24q^{157}+6q^{156}-37q^{155}-51q^{154}-28q^{153}-16q^{152}+14q^{151}+34q^{150}+30q^{149}+31q^{148}+q^{147}-25q^{146}-21q^{145}-27q^{144}-14q^{143}+q^{142}+10q^{141}+31q^{140}+20q^{139}+3q^{138}+2q^{137}-11q^{136}-16q^{135}-16q^{134}-13q^{133}+7q^{132}+10q^{131}+6q^{130}+16q^{129}+7q^{128}+2q^{127}-7q^{126}-13q^{125}-4q^{124}-6q^{123}-8q^{122}+5q^{121}+8q^{120}+8q^{119}+3q^{118}+6q^{116}-4q^{115}-9q^{114}-3q^{113}-q^{112}-q^{111}-2q^{110}+q^{109}+9q^{108}+2q^{107}-q^{106}+2q^{105}+q^{104}-q^{103}-6q^{102}-4q^{101}+4q^{100}-2q^{99}-2q^{98}+2q^{97}+3q^{96}+3q^{95}-q^{94}-2q^{93}+4q^{92}-2q^{91}-4q^{90}-q^{89}-q^{88}+2q^{87}-q^{86}-2q^{85}+4q^{84}+q^{83}-2q^{82}+q^{81}-q^{80}+2q^{79}-q^{78}-2q^{77}+q^{76}-2q^{74}+q^{73}-q^{72}+2q^{71}+q^{70}+q^{68}-2q^{66}+q^{65}-q^{64}+q^{60}-2q^{58}+q^{57}+q^{52}-2q^{50}+q^{44}+q^{36}+q^{28}$

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[10, 139]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[11, 19, 12, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8], X[13, 1, 14, 20], X[19, 13, 20, 12], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 139]]
Out[3]=	GaussCode[1, -10, 2, -1, -4, 6, -5, 7, 10, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8]
In[4]:=	DTCode[Knot[10, 139]]
Out[4]=	DTCode[4, 10, -14, -16, 2, -18, -20, -6, -8, -12]
In[5]:=	br = BR[Knot[10, 139]]
Out[5]=	BR[3, {1, 1, 1, 1, 2, 1, 1, 1, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 139]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 139]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 139]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 4, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 139]][t]
Out[10]=	-4 -3 2 3 4 -3 + t - t + - + 2 t - t + t t
In[11]:=	Conway[Knot[10, 139]][z]
Out[11]=	2 4 6 8 1 + 9 z + 14 z + 7 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 139]}
In[13]:=	{KnotDet[Knot[10, 139]], KnotSignature[Knot[10, 139]]}
Out[13]=	{3, 6}
In[14]:=	Jones[Knot[10, 139]][q]
Out[14]=	4 6 8 9 10 11 12 q + q - q + q - q + q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 139]}
In[16]:=	A2Invariant[Knot[10, 139]][q]
Out[16]=	14 16 18 20 22 28 32 34 36 38 40 q + q + 2 q + 2 q + q - q - q - q - q - q + q
In[17]:=	HOMFLYPT[Knot[10, 139]][a, z]
Out[17]=	2 2 2 4 4 6 6 8 -12 6 6 z 13 z 21 z 7 z 21 z z 8 z z a - --- + -- + --- - ----- + ----- - ---- + ----- - --- + ---- + -- 10 8 12 10 8 10 8 10 8 8 a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 139]][a, z]
Out[18]=	2 2 2 3 -12 6 6 2 z z 5 z 6 z 2 z 19 z 21 z z a + --- + -- - --- - --- - --- - --- - ---- - ----- - ----- + --- + 10 8 15 13 11 9 14 10 8 15 a a a a a a a a a a 3 3 3 4 4 4 5 5 6 z 13 z 13 z z 20 z 21 z 7 z 7 z 8 z --- + ----- + ----- + --- + ----- + ----- - ---- - ---- - ---- - 13 11 9 14 10 8 11 9 10 a a a a a a a a a 6 7 7 8 8 8 z z z z z ---- + --- + -- + --- + -- 8 11 9 10 8 a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 139]], Vassiliev[3][Knot[10, 139]]}
Out[19]=	{9, 25}
In[20]:=	Kh[Knot[10, 139]][q, t]
Out[20]=	7 9 11 2 15 3 13 4 15 4 15 5 17 5 q + q + q t + q t + q t + q t + q t + q t + 19 5 17 6 19 6 21 7 21 8 25 9 q t + q t + q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 139], 2][q]
Out[21]=	8 11 14 15 17 18 19 20 22 23 q + q + q - 2 q + 2 q - 2 q - q + 3 q - 3 q + 2 q + 24 25 26 27 28 29 30 31 32 33 q - 4 q + 2 q + q - 3 q + q + 2 q - q - q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

10_138 10_140