10 155

10_154

10_156

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10 155 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X7,16,8,17 X3,11,4,10 X15,3,16,2 X5,15,6,14 X11,5,12,4 X9,18,10,19 X20,14,1,13 X17,8,18,9 X12,20,13,19
Gauss code	-1, 4, -3, 6, -5, 1, -2, 9, -7, 3, -6, -10, 8, 5, -4, 2, -9, 7, 10, -8
Dowker-Thistlethwaite code	6 10 14 16 18 4 -20 2 8 -12
Conway Notation	[-3:2:2]

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 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-3][-7] Hyperbolic Volume 9.25054 A-Polynomial See Data:10 155/A-polynomial

[edit Notes for 10 155's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for 10 155's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-5t+7-5t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{5,t+1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$q^{6}-2q^{5}+3q^{4}-4q^{3}+4q^{2}-4q+4-2q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-5z^{4}a^{-2}+z^{4}a^{-4}+z^{4}-8z^{2}a^{-2}+3z^{2}a^{-4}+3z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}a^{-4}+z^{7}a^{-1}+3z^{7}a^{-3}+2z^{7}a^{-5}-3z^{6}a^{-2}-2z^{6}a^{-4}+z^{6}a^{-6}-z^{5}a^{-1}-9z^{5}a^{-3}-8z^{5}a^{-5}+7z^{4}a^{-2}-z^{4}a^{-4}-4z^{4}a^{-6}+4z^{4}+2az^{3}+6z^{3}a^{-3}+8z^{3}a^{-5}+a^{2}z^{2}-11z^{2}a^{-2}-z^{2}a^{-4}+4z^{2}a^{-6}-5z^{2}-2za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$q^{6}+2q^{2}+1-2q^{-6}-q^{-10}+q^{-14}+q^{-18}$
The G2 invariant	$q^{38}-q^{36}+q^{34}-q^{32}+q^{28}-2q^{26}+2q^{24}-q^{18}+q^{16}-q^{14}+2q^{12}+3q^{10}-2q^{8}+7q^{6}-6q^{4}+5q^{2}+5-8q^{-2}+13q^{-4}-9q^{-6}+2q^{-8}+7q^{-10}-11q^{-12}+9q^{-14}-4q^{-16}-5q^{-18}+7q^{-20}-10q^{-22}+2q^{-24}+2q^{-26}-11q^{-28}+10q^{-30}-11q^{-32}+4q^{-34}+q^{-36}-7q^{-38}+10q^{-40}-12q^{-42}+11q^{-44}-5q^{-46}-2q^{-48}+9q^{-50}-11q^{-52}+11q^{-54}-q^{-56}-5q^{-58}+10q^{-60}-8q^{-62}+3q^{-64}+7q^{-66}-12q^{-68}+12q^{-70}-6q^{-72}-2q^{-74}+9q^{-76}-11q^{-78}+10q^{-80}-5q^{-82}-q^{-84}+3q^{-86}-5q^{-88}+3q^{-90}-q^{-92}+q^{-94}$

Further Quantum Invariants

Further quantum knot invariants for 10_155.

A1 Invariants.

Weight	Invariant
1	$q^{5}-q^{3}+2q-q^{-7}+q^{-9}-q^{-11}+q^{-13}$
2	$q^{14}+q^{10}-4q^{6}+2q^{4}+3q^{2}-3+q^{-2}+3q^{-4}-q^{-6}-3q^{-8}+2q^{-10}+2q^{-12}-2q^{-14}+q^{-16}+3q^{-18}-2q^{-20}-3q^{-22}+3q^{-24}-4q^{-28}+2q^{-30}+3q^{-32}-2q^{-34}-q^{-36}+q^{-38}$
3	$2q^{27}-q^{23}-3q^{21}+6q^{17}+3q^{15}-7q^{13}-10q^{11}+4q^{9}+15q^{7}-q^{5}-14q^{3}-4q+13q^{-1}+11q^{-3}-5q^{-5}-9q^{-7}-2q^{-9}+8q^{-11}+5q^{-13}-5q^{-15}-9q^{-17}+5q^{-19}+9q^{-21}-5q^{-23}-10q^{-25}+4q^{-27}+11q^{-29}-3q^{-31}-12q^{-33}+12q^{-37}+4q^{-39}-9q^{-41}-9q^{-43}+6q^{-45}+12q^{-47}+q^{-49}-11q^{-51}-7q^{-53}+7q^{-55}+9q^{-57}-q^{-59}-8q^{-61}-4q^{-63}+4q^{-65}+4q^{-67}-2q^{-71}-q^{-73}+q^{-75}$
5	$q^{69}+q^{67}+q^{65}-3q^{63}-q^{61}+q^{59}-4q^{53}-8q^{51}+2q^{49}+20q^{47}+24q^{45}+2q^{43}-38q^{41}-64q^{39}-36q^{37}+54q^{35}+128q^{33}+104q^{31}-33q^{29}-179q^{27}-201q^{25}-41q^{23}+193q^{21}+293q^{19}+147q^{17}-145q^{15}-329q^{13}-255q^{11}+33q^{9}+299q^{7}+315q^{5}+81q^{3}-198q-300q^{-1}-166q^{-3}+74q^{-5}+233q^{-7}+199q^{-9}+28q^{-11}-130q^{-13}-174q^{-15}-96q^{-17}+34q^{-19}+126q^{-21}+116q^{-23}+26q^{-25}-70q^{-27}-108q^{-29}-57q^{-31}+45q^{-33}+98q^{-35}+54q^{-37}-40q^{-39}-92q^{-41}-43q^{-43}+59q^{-45}+102q^{-47}+30q^{-49}-88q^{-51}-130q^{-53}-33q^{-55}+113q^{-57}+164q^{-59}+56q^{-61}-125q^{-63}-203q^{-65}-98q^{-67}+112q^{-69}+235q^{-71}+155q^{-73}-65q^{-75}-241q^{-77}-215q^{-79}-7q^{-81}+210q^{-83}+258q^{-85}+96q^{-87}-136q^{-89}-262q^{-91}-184q^{-93}+33q^{-95}+215q^{-97}+228q^{-99}+80q^{-101}-115q^{-103}-219q^{-105}-162q^{-107}+148q^{-111}+182q^{-113}+97q^{-115}-41q^{-117}-142q^{-119}-142q^{-121}-52q^{-123}+58q^{-125}+118q^{-127}+102q^{-129}+27q^{-131}-57q^{-133}-94q^{-135}-68q^{-137}-7q^{-139}+45q^{-141}+66q^{-143}+44q^{-145}-34q^{-149}-40q^{-151}-24q^{-153}-q^{-155}+21q^{-157}+23q^{-159}+11q^{-161}-2q^{-163}-8q^{-165}-10q^{-167}-6q^{-169}+2q^{-171}+4q^{-173}+3q^{-175}+q^{-177}-2q^{-181}-q^{-183}+q^{-185}$
6	$q^{96}+2q^{94}-q^{90}-4q^{88}-q^{86}-q^{84}-2q^{82}+4q^{80}+4q^{78}+6q^{76}+2q^{74}+4q^{72}-9q^{70}-29q^{68}-29q^{66}-12q^{64}+32q^{62}+83q^{60}+114q^{58}+49q^{56}-95q^{54}-227q^{52}-256q^{50}-109q^{48}+188q^{46}+482q^{44}+502q^{42}+168q^{40}-369q^{38}-799q^{36}-788q^{34}-229q^{32}+614q^{30}+1163q^{28}+1022q^{26}+212q^{24}-828q^{22}-1450q^{20}-1183q^{18}-138q^{16}+1006q^{14}+1553q^{12}+1168q^{10}+86q^{8}-1044q^{6}-1503q^{4}-1040q^{2}-18+935q^{-2}+1286q^{-4}+885q^{-6}+14q^{-8}-754q^{-10}-1008q^{-12}-688q^{-14}-56q^{-16}+526q^{-18}+750q^{-20}+542q^{-22}+87q^{-24}-347q^{-26}-534q^{-28}-430q^{-30}-103q^{-32}+242q^{-34}+413q^{-36}+319q^{-38}+43q^{-40}-220q^{-42}-316q^{-44}-177q^{-46}+74q^{-48}+252q^{-50}+197q^{-52}-34q^{-54}-236q^{-56}-239q^{-58}-5q^{-60}+273q^{-62}+369q^{-64}+137q^{-66}-263q^{-68}-508q^{-70}-388q^{-72}+71q^{-74}+541q^{-76}+678q^{-78}+306q^{-80}-344q^{-82}-800q^{-84}-729q^{-86}-134q^{-88}+604q^{-90}+994q^{-92}+713q^{-94}-75q^{-96}-842q^{-98}-1092q^{-100}-645q^{-102}+234q^{-104}+1010q^{-106}+1152q^{-108}+565q^{-110}-363q^{-112}-1079q^{-114}-1155q^{-116}-538q^{-118}+413q^{-120}+1099q^{-122}+1125q^{-124}+518q^{-126}-363q^{-128}-1030q^{-130}-1097q^{-132}-540q^{-134}+276q^{-136}+895q^{-138}+1011q^{-140}+596q^{-142}-124q^{-144}-727q^{-146}-902q^{-148}-613q^{-150}-46q^{-152}+500q^{-154}+766q^{-156}+614q^{-158}+187q^{-160}-282q^{-162}-572q^{-164}-569q^{-166}-303q^{-168}+90q^{-170}+378q^{-172}+468q^{-174}+336q^{-176}+77q^{-178}-194q^{-180}-346q^{-182}-310q^{-184}-159q^{-186}+48q^{-188}+196q^{-190}+252q^{-192}+179q^{-194}+39q^{-196}-82q^{-198}-156q^{-200}-148q^{-202}-88q^{-204}+14q^{-206}+76q^{-208}+95q^{-210}+75q^{-212}+29q^{-214}-17q^{-216}-56q^{-218}-46q^{-220}-30q^{-222}-5q^{-224}+15q^{-226}+25q^{-228}+22q^{-230}+5q^{-232}-8q^{-236}-9q^{-238}-8q^{-240}+4q^{-244}+q^{-246}+3q^{-248}+q^{-250}-2q^{-254}-q^{-256}+q^{-258}$

A2 Invariants.

Weight	Invariant
1,0	$q^{6}+2q^{2}+1-2q^{-6}-q^{-10}+q^{-14}+q^{-18}$
1,1	$q^{20}-2q^{18}+4q^{16}-2q^{14}+5q^{12}-2q^{10}-2q^{8}+2q^{6}+2q^{2}+8-16q^{-2}+22q^{-4}-30q^{-6}+22q^{-8}-22q^{-10}+8q^{-12}+8q^{-14}-14q^{-16}+34q^{-18}-34q^{-20}+46q^{-22}-44q^{-24}+40q^{-26}-39q^{-28}+20q^{-30}-10q^{-32}-8q^{-34}+19q^{-36}-26q^{-38}+32q^{-40}-24q^{-42}+19q^{-44}-14q^{-46}+6q^{-48}-2q^{-50}+q^{-52}$
2,0	$q^{16}+q^{14}+3q^{12}+q^{10}-q^{8}-q^{2}-3-q^{-2}+q^{-4}-q^{-6}-q^{-8}+q^{-10}+2q^{-12}+q^{-14}+3q^{-16}+2q^{-18}+2q^{-20}+q^{-22}-q^{-26}-4q^{-28}-q^{-30}-q^{-34}-q^{-36}+q^{-38}+2q^{-40}-q^{-44}+q^{-48}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{16}-q^{14}-q^{12}+2q^{10}+q^{8}-q^{6}+4q^{4}+3q^{2}+3q^{-2}-3q^{-6}-2q^{-8}-3q^{-10}-2q^{-12}-3q^{-14}+3q^{-18}+q^{-20}+2q^{-22}+4q^{-24}-q^{-28}+q^{-30}-3q^{-32}+q^{-36}-q^{-38}+q^{-40}$
1,0,0	$q^{7}+3q^{3}+q+2q^{-1}-q^{-5}-2q^{-7}-2q^{-9}-q^{-11}-q^{-13}+q^{-15}+2q^{-19}+q^{-23}$
1,0,1	$q^{26}-2q^{24}+q^{22}+3q^{20}-2q^{18}+6q^{16}-2q^{14}+2q^{12}-2q^{10}+3q^{8}+3q^{6}+4q^{4}+15q^{2}-8+14q^{-2}-25q^{-4}+2q^{-6}-22q^{-8}-12q^{-10}+9q^{-12}-18q^{-14}+37q^{-16}-15q^{-18}+35q^{-20}-11q^{-22}+13q^{-24}+4q^{-26}-15q^{-28}+21q^{-30}-27q^{-32}+22q^{-34}-25q^{-36}+12q^{-38}-14q^{-40}-3q^{-42}+8q^{-44}-14q^{-46}+21q^{-48}-13q^{-50}+15q^{-52}-5q^{-54}-2q^{-56}+4q^{-58}-7q^{-60}+5q^{-62}-2q^{-64}+q^{-66}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{18}-q^{14}+q^{12}+3q^{10}-q^{8}+2q^{6}+6q^{4}+5q^{2}+1+2q^{-2}+2q^{-4}-4q^{-6}-6q^{-8}-3q^{-10}-4q^{-12}-8q^{-14}-q^{-16}+q^{-18}-2q^{-20}+3q^{-22}+8q^{-24}+5q^{-26}+2q^{-28}+3q^{-30}+2q^{-32}-3q^{-34}-4q^{-36}-q^{-38}-q^{-40}-q^{-42}+q^{-44}+q^{-46}+q^{-50}$
1,0,0,0	$q^{8}+3q^{4}+2q^{2}+2+2q^{-2}-q^{-6}-3q^{-8}-2q^{-10}-3q^{-12}-q^{-14}-q^{-16}+q^{-18}+q^{-20}+q^{-22}+2q^{-24}+q^{-28}$

B2 Invariants.

Weight	Invariant
0,1	$q^{16}-q^{14}+q^{12}-2q^{10}+3q^{8}-3q^{6}+6q^{4}-q^{2}+2+q^{-2}+q^{-6}-6q^{-8}+5q^{-10}-8q^{-12}+5q^{-14}-6q^{-16}+5q^{-18}-3q^{-20}+2q^{-22}+2q^{-24}-2q^{-26}+3q^{-28}-3q^{-30}+3q^{-32}-4q^{-34}+3q^{-36}-q^{-38}+q^{-40}$
1,0	$q^{26}-q^{22}-q^{20}+2q^{16}+2q^{14}-2q^{10}-q^{8}+5q^{6}+2q^{4}-q^{2}-2+2q^{-2}+3q^{-4}-4q^{-8}-2q^{-10}+3q^{-12}-3q^{-16}-3q^{-18}+q^{-20}+q^{-22}-q^{-24}-2q^{-26}+q^{-28}+3q^{-30}+q^{-32}-2q^{-34}-q^{-36}+4q^{-38}+4q^{-40}-q^{-42}-4q^{-44}+3q^{-48}+q^{-50}-3q^{-52}-3q^{-54}+2q^{-56}+2q^{-58}-q^{-60}-q^{-62}+q^{-66}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{22}-q^{20}-q^{16}+2q^{14}-q^{12}+2q^{10}-2q^{8}+6q^{6}+5q^{2}+2+3q^{-2}+q^{-4}-6q^{-10}-9q^{-14}+q^{-16}-8q^{-18}+3q^{-20}-5q^{-22}+5q^{-24}-q^{-26}+6q^{-28}+2q^{-30}+3q^{-32}+2q^{-34}-q^{-36}+3q^{-38}-3q^{-40}+q^{-42}-4q^{-44}+3q^{-46}-3q^{-48}+2q^{-50}-q^{-52}+q^{-54}$

G2 Invariants.

Weight

Invariant

1,0

$q^{38}-q^{36}+q^{34}-q^{32}+q^{28}-2q^{26}+2q^{24}-q^{18}+q^{16}-q^{14}+2q^{12}+3q^{10}-2q^{8}+7q^{6}-6q^{4}+5q^{2}+5-8q^{-2}+13q^{-4}-9q^{-6}+2q^{-8}+7q^{-10}-11q^{-12}+9q^{-14}-4q^{-16}-5q^{-18}+7q^{-20}-10q^{-22}+2q^{-24}+2q^{-26}-11q^{-28}+10q^{-30}-11q^{-32}+4q^{-34}+q^{-36}-7q^{-38}+10q^{-40}-12q^{-42}+11q^{-44}-5q^{-46}-2q^{-48}+9q^{-50}-11q^{-52}+11q^{-54}-q^{-56}-5q^{-58}+10q^{-60}-8q^{-62}+3q^{-64}+7q^{-66}-12q^{-68}+12q^{-70}-6q^{-72}-2q^{-74}+9q^{-76}-11q^{-78}+10q^{-80}-5q^{-82}-q^{-84}+3q^{-86}-5q^{-88}+3q^{-90}-q^{-92}+q^{-94}$

.

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["10 155"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-3z^{4}-2z^{2}+1$

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]=

$\{5,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}a^{-4}+z^{7}a^{-1}+3z^{7}a^{-3}+2z^{7}a^{-5}-3z^{6}a^{-2}-2z^{6}a^{-4}+z^{6}a^{-6}-z^{5}a^{-1}-9z^{5}a^{-3}-8z^{5}a^{-5}+7z^{4}a^{-2}-z^{4}a^{-4}-4z^{4}a^{-6}+4z^{4}+2az^{3}+6z^{3}a^{-3}+8z^{3}a^{-5}+a^{2}z^{2}-11z^{2}a^{-2}-z^{2}a^{-4}+4z^{2}a^{-6}-5z^{2}-2za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_9, K11n37, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_137, K11n37, ...}

Vassiliev invariants

V2 and V3:

(-2, -2)

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 $-8$ $-16$ $32$ ${\frac {164}{3}}$ ${\frac {124}{3}}$ $128$ ${\frac {800}{3}}$ ${\frac {224}{3}}$ $80$ $-{\frac {256}{3}}$ $128$ $-{\frac {1312}{3}}$ $-{\frac {992}{3}}$ $-{\frac {31}{15}}$ ${\frac {4924}{15}}$ $-{\frac {25564}{45}}$ ${\frac {1279}{9}}$ $-{\frac {2911}{15}}$

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=$0 is the signature of 10 155. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{18}-2q^{17}-q^{16}+6q^{15}-3q^{14}-7q^{13}+10q^{12}-13q^{10}+11q^{9}+5q^{8}-15q^{7}+8q^{6}+9q^{5}-15q^{4}+3q^{3}+11q^{2}-11q+1+7q^{-1}-5q^{-2}+q^{-4}-q^{-5}+q^{-6}$
3	$q^{36}-2q^{35}-q^{34}+2q^{33}+5q^{32}-2q^{31}-9q^{30}-2q^{29}+12q^{28}+8q^{27}-11q^{26}-16q^{25}+8q^{24}+20q^{23}-22q^{21}-7q^{20}+20q^{19}+13q^{18}-14q^{17}-19q^{16}+8q^{15}+22q^{14}-26q^{12}-6q^{11}+27q^{10}+14q^{9}-30q^{8}-20q^{7}+31q^{6}+24q^{5}-27q^{4}-30q^{3}+24q^{2}+28q-11-28q^{-1}+7q^{-2}+18q^{-3}+2q^{-4}-12q^{-5}-4q^{-6}+4q^{-7}+5q^{-8}-2q^{-9}-q^{-10}-2q^{-11}+2q^{-12}$
4	$q^{60}-2q^{59}-q^{58}+2q^{57}+q^{56}+6q^{55}-6q^{54}-7q^{53}-2q^{52}-2q^{51}+23q^{50}+3q^{49}-7q^{48}-15q^{47}-28q^{46}+25q^{45}+19q^{44}+22q^{43}-2q^{42}-57q^{41}-5q^{40}+q^{39}+45q^{38}+44q^{37}-41q^{36}-23q^{35}-51q^{34}+18q^{33}+73q^{32}+9q^{31}+8q^{30}-85q^{29}-43q^{28}+56q^{27}+50q^{26}+67q^{25}-81q^{24}-97q^{23}+14q^{22}+66q^{21}+120q^{20}-60q^{19}-134q^{18}-26q^{17}+77q^{16}+158q^{15}-43q^{14}-163q^{13}-58q^{12}+88q^{11}+188q^{10}-22q^{9}-185q^{8}-92q^{7}+81q^{6}+204q^{5}+17q^{4}-169q^{3}-119q^{2}+35q+178+60q^{-1}-101q^{-2}-106q^{-3}-19q^{-4}+100q^{-5}+61q^{-6}-27q^{-7}-51q^{-8}-33q^{-9}+30q^{-10}+28q^{-11}+q^{-12}-9q^{-13}-15q^{-14}+5q^{-15}+5q^{-16}+q^{-17}-4q^{-19}+q^{-20}+q^{-21}$
5	$q^{90}-2q^{89}-q^{88}+2q^{87}+q^{86}+2q^{85}+2q^{84}-4q^{83}-9q^{82}-2q^{81}+3q^{80}+8q^{79}+15q^{78}+8q^{77}-11q^{76}-24q^{75}-20q^{74}-8q^{73}+21q^{72}+42q^{71}+33q^{70}-2q^{69}-41q^{68}-60q^{67}-40q^{66}+16q^{65}+70q^{64}+82q^{63}+34q^{62}-44q^{61}-100q^{60}-94q^{59}-20q^{58}+82q^{57}+135q^{56}+94q^{55}-15q^{54}-128q^{53}-168q^{52}-80q^{51}+78q^{50}+198q^{49}+180q^{48}+20q^{47}-181q^{46}-262q^{45}-139q^{44}+120q^{43}+306q^{42}+252q^{41}-19q^{40}-310q^{39}-356q^{38}-88q^{37}+280q^{36}+428q^{35}+201q^{34}-230q^{33}-479q^{32}-298q^{31}+175q^{30}+506q^{29}+382q^{28}-122q^{27}-530q^{26}-444q^{25}+78q^{24}+548q^{23}+500q^{22}-50q^{21}-573q^{20}-546q^{19}+29q^{18}+600q^{17}+594q^{16}-6q^{15}-626q^{14}-648q^{13}-22q^{12}+638q^{11}+690q^{10}+84q^{9}-616q^{8}-740q^{7}-152q^{6}+560q^{5}+734q^{4}+242q^{3}-445q^{2}-706q-311+320q^{-1}+600q^{-2}+344q^{-3}-166q^{-4}-472q^{-5}-327q^{-6}+54q^{-7}+312q^{-8}+270q^{-9}+18q^{-10}-180q^{-11}-181q^{-12}-46q^{-13}+78q^{-14}+110q^{-15}+40q^{-16}-34q^{-17}-44q^{-18}-22q^{-19}+4q^{-20}+20q^{-21}+12q^{-22}-8q^{-23}-4q^{-24}+2q^{-27}+2q^{-28}-4q^{-29}+q^{-32}$
6	$q^{126}-2q^{125}-q^{124}+2q^{123}+q^{122}+2q^{121}-2q^{120}+4q^{119}-6q^{118}-9q^{117}+q^{116}+2q^{115}+10q^{114}+3q^{113}+21q^{112}-3q^{111}-19q^{110}-19q^{109}-23q^{108}-6q^{107}-7q^{106}+60q^{105}+43q^{104}+27q^{103}+q^{102}-42q^{101}-68q^{100}-109q^{99}+35q^{97}+101q^{96}+122q^{95}+98q^{94}+5q^{93}-165q^{92}-148q^{91}-172q^{90}-50q^{89}+86q^{88}+250q^{87}+276q^{86}+94q^{85}-16q^{84}-262q^{83}-338q^{82}-307q^{81}-16q^{80}+273q^{79}+384q^{78}+453q^{77}+165q^{76}-186q^{75}-573q^{74}-562q^{73}-294q^{72}+95q^{71}+628q^{70}+768q^{69}+534q^{68}-158q^{67}-678q^{66}-913q^{65}-721q^{64}+71q^{63}+835q^{62}+1201q^{61}+723q^{60}-71q^{59}-939q^{58}-1407q^{57}-880q^{56}+218q^{55}+1277q^{54}+1439q^{53}+857q^{52}-352q^{51}-1549q^{50}-1656q^{49}-661q^{48}+830q^{47}+1689q^{46}+1624q^{45}+436q^{44}-1268q^{43}-2046q^{42}-1399q^{41}+235q^{40}+1618q^{39}+2080q^{38}+1086q^{37}-896q^{36}-2183q^{35}-1869q^{34}-224q^{33}+1498q^{32}+2325q^{31}+1486q^{30}-664q^{29}-2279q^{28}-2147q^{27}-458q^{26}+1501q^{25}+2527q^{24}+1717q^{23}-609q^{22}-2457q^{21}-2398q^{20}-597q^{19}+1597q^{18}+2790q^{17}+1993q^{16}-515q^{15}-2628q^{14}-2743q^{13}-924q^{12}+1493q^{11}+2977q^{10}+2427q^{9}-60q^{8}-2420q^{7}-2967q^{6}-1506q^{5}+861q^{4}+2657q^{3}+2681q^{2}+708q-1549-2566q^{-1}-1857q^{-2}-92q^{-3}+1635q^{-4}+2218q^{-5}+1167q^{-6}-419q^{-7}-1484q^{-8}-1472q^{-9}-639q^{-10}+491q^{-11}+1173q^{-12}+900q^{-13}+203q^{-14}-444q^{-15}-662q^{-16}-498q^{-17}-58q^{-18}+330q^{-19}+341q^{-20}+192q^{-21}-14q^{-22}-125q^{-23}-164q^{-24}-78q^{-25}+36q^{-26}+44q^{-27}+45q^{-28}+15q^{-29}+7q^{-30}-20q^{-31}-13q^{-32}+5q^{-33}-7q^{-34}+q^{-35}-2q^{-36}+7q^{-37}+3q^{-40}-3q^{-41}-q^{-42}-2q^{-43}+q^{-44}+q^{-45}$
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[10, 155]]
Out[2]=	PD[X[1, 6, 2, 7], X[7, 16, 8, 17], X[3, 11, 4, 10], X[15, 3, 16, 2], X[5, 15, 6, 14], X[11, 5, 12, 4], X[9, 18, 10, 19], X[20, 14, 1, 13], X[17, 8, 18, 9], X[12, 20, 13, 19]]
In[3]:=	GaussCode[Knot[10, 155]]
Out[3]=	GaussCode[-1, 4, -3, 6, -5, 1, -2, 9, -7, 3, -6, -10, 8, 5, -4, 2, -9, 7, 10, -8]
In[4]:=	DTCode[Knot[10, 155]]
Out[4]=	DTCode[6, 10, 14, 16, 18, 4, -20, 2, 8, -12]
In[5]:=	br = BR[Knot[10, 155]]
Out[5]=	BR[3, {1, 1, 1, 2, -1, -1, 2, -1, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 155]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 155]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 155]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 155]][t]
Out[10]=	-3 3 5 2 3 7 - t + -- - - - 5 t + 3 t - t 2 t t
In[11]:=	Conway[Knot[10, 155]][z]
Out[11]=	2 4 6 1 - 2 z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 9], Knot[10, 155], Knot[11, NonAlternating, 37]}
In[13]:=	{KnotDet[Knot[10, 155]], KnotSignature[Knot[10, 155]]}
Out[13]=	{25, 0}
In[14]:=	Jones[Knot[10, 155]][q]
Out[14]=	-2 2 2 3 4 5 6 4 + q - - - 4 q + 4 q - 4 q + 3 q - 2 q + q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 137], Knot[10, 155], Knot[11, NonAlternating, 37]}
In[16]:=	A2Invariant[Knot[10, 155]][q]
Out[16]=	-6 2 6 10 14 18 1 + q + -- - 2 q - q + q + q 2 q
In[17]:=	HOMFLYPT[Knot[10, 155]][a, z]
Out[17]=	2 2 4 4 6 2 4 2 3 z 8 z 4 z 5 z z 3 + -- - -- + 3 z + ---- - ---- + z + -- - ---- - -- 4 2 4 2 4 2 2 a a a a a a a
In[18]:=	Kauffman[Knot[10, 155]][a, z]
Out[18]=	2 2 2 3 2 4 2 z 2 z 2 4 z z 11 z 2 2 8 z 3 + -- + -- - --- - --- - 5 z + ---- - -- - ----- + a z + ---- + 4 2 5 3 6 4 2 5 a a a a a a a a 3 4 4 4 5 5 5 6 6 z 3 4 4 z z 7 z 8 z 9 z z z ---- + 2 a z + 4 z - ---- - -- + ---- - ---- - ---- - -- + -- - 3 6 4 2 5 3 a 6 a a a a a a a 6 6 7 7 7 8 8 2 z 3 z 2 z 3 z z z z ---- - ---- + ---- + ---- + -- + -- + -- 4 2 5 3 a 4 2 a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 155]], Vassiliev[3][Knot[10, 155]]}
Out[19]=	{-2, -2}
In[20]:=	Kh[Knot[10, 155]][q, t]
Out[20]=	3 1 1 1 3 3 2 5 2 - + 2 q + ----- + ---- + --- + 2 q t + 2 q t + 2 q t + 2 q t + q 5 2 3 q t q t q t 5 3 7 3 7 4 9 4 9 5 11 5 13 6 2 q t + 2 q t + q t + 2 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 155], 2][q]
Out[21]=	-6 -5 -4 5 7 2 3 4 5 1 + q - q + q - -- + - - 11 q + 11 q + 3 q - 15 q + 9 q + 2 q q 6 7 8 9 10 12 13 14 8 q - 15 q + 5 q + 11 q - 13 q + 10 q - 7 q - 3 q + 15 16 17 18 6 q - q - 2 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

10_154 10_156