© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:

10112 10114

KnotPlot

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KnotPlot

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X12,7,13,8 X8,18,9,17 X6,19,7,20 X16,12,17,11 X18,13,19,14 X2,10,3,9

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 12 2 16 18 20 8 6

Minimum Braid Representative: Length is 11, width is 4 Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 1 3 3 / NotAvailable 1

Alexander Polynomial:

2t-3 - 11t-2 + 26t-1 - 33 + 26t - 11t2 + 2t3

Conway Polynomial:

1 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a107, K11a347, ...}

Determinant and Signature:

{111, 2}

Jones Polynomial:

- q-2 + 4q-1 - 8 + 14q - 17q2 + 19q3 - 18q4 + 14q5 - 10q6 + 5q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 1 + 5q2 - 2q4 + 4q6 - 2q10 + q12 - 5q14 + 3q16 - q18 - q20 + 3q22 - q24

HOMFLY-PT Polynomial:

a-6 - a-6z4 - 3a-4 - 2a-4z2 + a-4z4 + a-4z6 + 3a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

a-9z5 - 5a-8z4 + 5a-8z6 + a-7z + 5a-7z3 - 16a-7z5 + 10a-7z7 - a-6 + 3a-6z2 - 4a-6z4 - 9a-6z6 + 9a-6z8 + a-5z + 16a-5z3 - 36a-5z5 + 15a-5z7 + 3a-5z9 - 3a-4 + 8a-4z2 + a-4z4 - 23a-4z6 + 16a-4z8 - a-3z + 17a-3z3 - 30a-3z5 + 12a-3z7 + 3a-3z9 - 3a-2 + 8a-2z2 - 6a-2z4 - 5a-2z6 + 7a-2z8 - a-1z + 5a-1z3 - 10a-1z5 + 7a-1z7 + 3z2 - 6z4 + 4z6 - az3 + az5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{0, -1}

Khovanov Homology: (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 10113. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                     1 j = 15                   4   j = 13                 6 1   j = 11               8 4     j = 9             10 6       j = 7           9 8         j = 5         8 10           j = 3       6 9             j = 1     3 9               j = -1   1 5                 j = -3   3                   j = -5 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-7 - 4q-6 + 3q-5 + 11q-4 - 29q-3 + 12q-2 + 56q-1 - 94 + 4q + 157q2 - 171q3 - 47q4 + 269q5 - 203q6 - 116q7 + 325q8 - 174q9 - 161q10 + 295q11 - 102q12 - 159q13 + 196q14 - 27q15 - 108q16 + 83q17 + 8q18 - 41q19 + 16q20 + 5q21 - 5q22 + q23
3	- q-15 + 4q-14 - 3q-13 - 6q-12 + 4q-11 + 20q-10 - 13q-9 - 54q-8 + 33q-7 + 118q-6 - 39q-5 - 252q-4 + 23q-3 + 449q-2 + 76q-1 - 715 - 268q + 974q2 + 617q3 - 1230q4 - 1024q5 + 1347q6 + 1525q7 - 1388q8 - 1976q9 + 1283q10 + 2385q11 - 1113q12 - 2660q13 + 853q14 + 2834q15 - 568q16 - 2867q17 + 255q18 + 2767q19 + 73q20 - 2554q21 - 364q22 + 2202q23 + 624q24 - 1784q25 - 767q26 + 1301q27 + 808q28 - 844q29 - 730q30 + 462q31 + 572q32 - 197q33 - 383q34 + 49q35 + 213q36 + 17q37 - 108q38 - 14q39 + 36q40 + 11q41 - 11q42 - 5q43 + 5q44 - q45
4	q-26 - 4q-25 + 3q-24 + 6q-23 - 9q-22 + 5q-21 - 19q-20 + 26q-19 + 37q-18 - 70q-17 - 17q-16 - 69q-15 + 179q-14 + 248q-13 - 268q-12 - 330q-11 - 441q-10 + 662q-9 + 1302q-8 - 201q-7 - 1348q-6 - 2326q-5 + 829q-4 + 4049q-3 + 1895q-2 - 2153q-1 - 6996 - 1746q + 7305q2 + 7617q3 + 302q4 - 12951q5 - 8838q6 + 7485q7 + 15153q8 + 7772q9 - 16185q10 - 18207q11 + 2697q12 + 20332q13 + 17731q14 - 14666q15 - 25517q16 - 4872q17 + 21092q18 + 26018q19 - 9929q20 - 28602q21 - 11987q22 + 18367q23 + 30657q24 - 4111q25 - 27794q26 - 17291q27 + 13360q28 + 31572q29 + 2084q30 - 23517q31 - 20505q32 + 6390q33 + 28515q34 + 7975q35 - 15829q36 - 20470q37 - 1369q38 + 21118q39 + 11329q40 - 6381q41 - 15986q42 - 6767q43 + 11259q44 + 10069q45 + 897q46 - 8631q47 - 7254q48 + 3239q49 + 5529q50 + 3229q51 - 2542q52 - 4248q53 - 206q54 + 1563q55 + 2075q56 - 26q57 - 1402q58 - 469q59 + 48q60 + 645q61 + 212q62 - 251q63 - 112q64 - 82q65 + 103q66 + 54q67 - 30q68 - 6q69 - 16q70 + 11q71 + 5q72 - 5q73 + q74
5	- q-40 + 4q-39 - 3q-38 - 6q-37 + 9q-36 - 6q-34 + 6q-33 - 9q-32 - 15q-31 + 47q-30 + 40q-29 - 46q-28 - 98q-27 - 111q-26 + 26q-25 + 328q-24 + 435q-23 - 6q-22 - 799q-21 - 1165q-20 - 405q-19 + 1431q-18 + 2885q-17 + 1877q-16 - 2040q-15 - 5903q-14 - 5328q-13 + 1302q-12 + 10079q-11 + 12382q-10 + 2521q-9 - 14367q-8 - 23416q-7 - 12198q-6 + 15804q-5 + 38155q-4 + 29891q-3 - 10729q-2 - 53407q-1 - 56439 - 5068q + 64870q2 + 89488q3 + 33970q4 - 66442q5 - 124947q6 - 75683q7 + 54716q8 + 155697q9 + 126211q10 - 26835q11 - 176795q12 - 179900q13 - 13836q14 + 183631q15 + 229332q16 + 64083q17 - 176317q18 - 269958q19 - 116345q20 + 156560q21 + 298125q22 + 166215q23 - 128881q24 - 314188q25 - 208866q26 + 97142q27 + 319348q28 + 243562q29 - 65076q30 - 316379q31 - 269756q32 + 33780q33 + 307134q34 + 289392q35 - 3679q36 - 292904q37 - 303206q38 - 26342q39 + 273152q40 + 312043q41 + 57310q42 - 247013q43 - 315143q44 - 89127q45 + 213088q46 + 310323q47 + 120958q48 - 170921q49 - 295846q50 - 149290q51 + 122008q52 + 269064q53 + 170524q54 - 69435q55 - 230663q56 - 180144q57 + 18982q58 + 182272q59 + 175703q60 + 23740q61 - 129444q62 - 156999q63 - 53411q64 + 78290q65 + 127438q66 + 67642q67 - 35372q68 - 92461q69 - 67378q70 + 4708q71 + 58646q72 + 56673q73 + 12436q74 - 31019q75 - 40936q76 - 18233q77 + 11986q78 + 25488q79 + 16722q80 - 1683q81 - 13351q82 - 11815q83 - 2529q84 + 5594q85 + 7090q86 + 2924q87 - 1743q88 - 3380q89 - 2076q90 + 120q91 + 1425q92 + 1142q93 + 146q94 - 463q95 - 470q96 - 147q97 + 105q98 + 193q99 + 77q100 - 53q101 - 49q102 - 10q103 + 11q105 + 16q106 - 11q107 - 5q108 + 5q109 - q110

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 113]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], > X[12, 7, 13, 8], X[8, 18, 9, 17], X[6, 19, 7, 20], X[16, 12, 17, 11], > X[18, 13, 19, 14], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 113]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -7, 5, -6, 10, -2, 8, -5, 9, -3, 4, -8, 6, -9, 7, > -4]
In[4]:=	DTCode[Knot[10, 113]]
Out[4]=	DTCode[4, 10, 14, 12, 2, 16, 18, 20, 8, 6]
In[5]:=	br = BR[Knot[10, 113]]
Out[5]=	BR[4, {1, 1, 1, 2, -3, 2, -1, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 113]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 113]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 113]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 113]][t]
Out[10]=	2 11 26 2 3 -33 + -- - -- + -- + 26 t - 11 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 113]][z]
Out[11]=	4 6 1 + z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 113], Knot[11, Alternating, 107], Knot[11, Alternating, 347]}
In[13]:=	{KnotDet[Knot[10, 113]], KnotSignature[Knot[10, 113]]}
Out[13]=	{111, 2}
In[14]:=	Jones[Knot[10, 113]][q]
Out[14]=	-2 4 2 3 4 5 6 7 8 -8 - q + - + 14 q - 17 q + 19 q - 18 q + 14 q - 10 q + 5 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 113]}
In[16]:=	A2Invariant[Knot[10, 113]][q]
Out[16]=	-6 2 -2 2 4 6 10 12 14 16 18 -1 - q + -- - q + 5 q - 2 q + 4 q - 2 q + q - 5 q + 3 q - q - 4 q 20 22 24 > q + 3 q - q
In[17]:=	HOMFLYPT[Knot[10, 113]][a, z]
Out[17]=	2 2 4 4 4 6 6 -6 3 3 2 2 z 3 z 4 z z 2 z z z a - -- + -- - z - ---- + ---- - z - -- + -- + ---- + -- + -- 4 2 4 2 6 4 2 4 2 a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 113]][a, z]
Out[18]=	2 2 2 3 3 -6 3 3 z z z z 2 3 z 8 z 8 z 5 z 16 z -a - -- - -- + -- + -- - -- - - + 3 z + ---- + ---- + ---- + ---- + ----- + 4 2 7 5 3 a 6 4 2 7 5 a a a a a a a a a a 3 3 4 4 4 4 5 5 5 17 z 5 z 3 4 5 z 4 z z 6 z z 16 z 36 z > ----- + ---- - a z - 6 z - ---- - ---- + -- - ---- + -- - ----- - ----- - 3 a 8 6 4 2 9 7 5 a a a a a a a a 5 5 6 6 6 6 7 7 30 z 10 z 5 6 5 z 9 z 23 z 5 z 10 z 15 z > ----- - ----- + a z + 4 z + ---- - ---- - ----- - ---- + ----- + ----- + 3 a 8 6 4 2 7 5 a a a a a a a 7 7 8 8 8 9 9 12 z 7 z 9 z 16 z 7 z 3 z 3 z > ----- + ---- + ---- + ----- + ---- + ---- + ---- 3 a 6 4 2 5 3 a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 113]], Vassiliev[3][Knot[10, 113]]}
Out[19]=	{0, -1}
In[20]:=	Kh[Knot[10, 113]][q, t]
Out[20]=	3 1 3 1 5 3 q 3 5 5 2 9 q + 6 q + ----- + ----- + ---- + --- + --- + 9 q t + 8 q t + 10 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 9 q t + 8 q t + 10 q t + 6 q t + 8 q t + 4 q t + 6 q t + 13 6 15 6 17 7 > q t + 4 q t + q t
In[21]:=	ColouredJones[Knot[10, 113], 2][q]
Out[21]=	-7 4 3 11 29 12 56 2 3 4 -94 + q - -- + -- + -- - -- + -- + -- + 4 q + 157 q - 171 q - 47 q + 6 5 4 3 2 q q q q q q 5 6 7 8 9 10 11 12 > 269 q - 203 q - 116 q + 325 q - 174 q - 161 q + 295 q - 102 q - 13 14 15 16 17 18 19 20 > 159 q + 196 q - 27 q - 108 q + 83 q + 8 q - 41 q + 16 q + 21 22 23 > 5 q - 5 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10113

10112 10114