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936 938

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KnotPlot

PD Presentation:

X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 12 16 2 6 18 8

Minimum Braid Representative: Length is 12, width is 5 Braid index is 5

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2 2 3 / 4--7 2

Alexander Polynomial:

2t-2 - 11t-1 + 19 - 11t + 2t2

Conway Polynomial:

1 - 3z2 + 2z4

Other knots with the same Alexander/Conway Polynomial:

{K11n100, ...}

Determinant and Signature:

{45, 0}

Jones Polynomial:

- q-5 + 3q-4 - 4q-3 + 7q-2 - 8q-1 + 7 - 7q + 5q2 - 2q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + q-12 - q-10 + 3q-8 + q-6 - 3 - 2q4 + q6 + 2q8 - q10 + q12 + q14

HOMFLY-PT Polynomial:

a-4 - 2a-2z2 - 2 - z2 + z4 + 2a2 + a2z2 + a2z4 - a4z2

Kauffman Polynomial:

a-4 - 2a-4z2 + a-4z4 - 2a-3z3 + 2a-3z5 + a-2z2 - 3a-2z4 + 3a-2z6 - 5a-1z + 6a-1z3 - 4a-1z5 + 3a-1z7 - 2 + 12z2 - 13z4 + 5z6 + z8 - 7az + 13az3 - 13az5 + 6az7 - 2a2 + 14a2z2 - 17a2z4 + 5a2z6 + a2z8 - 2a3z + 3a3z3 - 6a3z5 + 3a3z7 + 5a4z2 - 8a4z4 + 3a4z6 - 2a5z3 + a5z5

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

{-3, -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=0 is the signature of 937. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   1 j = 7                 1   j = 5               4 1   j = 3             3 1     j = 1           4 4       j = -1         5 4         j = -3       2 3           j = -5     2 5             j = -7   1 2               j = -9   2                 j = -11 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-15 - 3q-14 + 9q-12 - 11q-11 - 5q-10 + 26q-9 - 18q-8 - 20q-7 + 46q-6 - 20q-5 - 39q-4 + 59q-3 - 14q-2 - 48q-1 + 58 - 5q - 44q2 + 41q3 + q4 - 29q5 + 19q6 + 3q7 - 11q8 + 5q9 + q10 - 2q11 + q12
3	- q-30 + 3q-29 - 5q-27 - 4q-26 + 11q-25 + 11q-24 - 19q-23 - 22q-22 + 23q-21 + 44q-20 - 26q-19 - 66q-18 + 14q-17 + 100q-16 - 2q-15 - 120q-14 - 34q-13 + 151q-12 + 57q-11 - 159q-10 - 100q-9 + 172q-8 + 132q-7 - 172q-6 - 158q-5 + 165q-4 + 183q-3 - 156q-2 - 189q-1 + 130 + 197q - 112q2 - 183q3 + 78q4 + 166q5 - 51q6 - 137q7 + 26q8 + 104q9 - 4q10 - 77q11 + q12 + 42q13 + 9q14 - 27q15 - 3q16 + 11q17 + 3q18 - 7q19 + q20 + q21 + q22 - 2q23 + q24
4	q-50 - 3q-49 + 5q-47 + 4q-45 - 18q-44 - 4q-43 + 20q-42 + 8q-41 + 25q-40 - 57q-39 - 37q-38 + 35q-37 + 39q-36 + 102q-35 - 101q-34 - 123q-33 - 8q-32 + 61q-31 + 275q-30 - 68q-29 - 221q-28 - 169q-27 - 25q-26 + 494q-25 + 107q-24 - 210q-23 - 396q-22 - 282q-21 + 632q-20 + 366q-19 - 32q-18 - 564q-17 - 637q-16 + 626q-15 + 596q-14 + 241q-13 - 626q-12 - 959q-11 + 523q-10 + 734q-9 + 492q-8 - 603q-7 - 1163q-6 + 373q-5 + 774q-4 + 680q-3 - 511q-2 - 1231q-1 + 197 + 705q + 780q2 - 336q3 - 1137q4 + 4q5 + 506q6 + 759q7 - 102q8 - 875q9 - 139q10 + 229q11 + 591q12 + 89q13 - 517q14 - 163q15 + 7q16 + 337q17 + 140q18 - 213q19 - 86q20 - 73q21 + 126q22 + 87q23 - 59q24 - 13q25 - 49q26 + 31q27 + 27q28 - 18q29 + 9q30 - 15q31 + 6q32 + 5q33 - 7q34 + 5q35 - 3q36 + q37 + q38 - 2q39 + q40
5	- q-75 + 3q-74 - 5q-72 + 3q-69 + 11q-68 + 4q-67 - 20q-66 - 18q-65 - 2q-64 + 18q-63 + 42q-62 + 29q-61 - 33q-60 - 84q-59 - 59q-58 + 28q-57 + 120q-56 + 141q-55 + 17q-54 - 177q-53 - 251q-52 - 101q-51 + 176q-50 + 378q-49 + 297q-48 - 112q-47 - 512q-46 - 531q-45 - 89q-44 + 538q-43 + 855q-42 + 425q-41 - 465q-40 - 1096q-39 - 909q-38 + 148q-37 + 1314q-36 + 1440q-35 + 304q-34 - 1260q-33 - 1997q-32 - 1003q-31 + 1124q-30 + 2436q-29 + 1697q-28 - 662q-27 - 2764q-26 - 2507q-25 + 182q-24 + 2928q-23 + 3153q-22 + 470q-21 - 2940q-20 - 3791q-19 - 1051q-18 + 2861q-17 + 4241q-16 + 1618q-15 - 2701q-14 - 4596q-13 - 2125q-12 + 2522q-11 + 4858q-10 + 2518q-9 - 2302q-8 - 4986q-7 - 2903q-6 + 2059q-5 + 5081q-4 + 3163q-3 - 1752q-2 - 4996q-1 - 3460 + 1391q + 4851q2 + 3609q3 - 933q4 - 4481q5 - 3755q6 + 425q7 + 4010q8 + 3701q9 + 110q10 - 3325q11 - 3544q12 - 607q13 + 2576q14 + 3185q15 + 982q16 - 1780q17 - 2660q18 - 1216q19 + 1032q20 + 2093q21 + 1226q22 - 460q23 - 1429q24 - 1104q25 + 16q26 + 926q27 + 860q28 + 155q29 - 454q30 - 596q31 - 251q32 + 211q33 + 356q34 + 191q35 - 38q36 - 184q37 - 146q38 + 6q39 + 72q40 + 67q41 + 30q42 - 31q43 - 39q44 - 3q45 + 5q46 + 2q47 + 17q48 - 2q49 - 10q50 + 4q51 - 5q53 + 5q54 + q55 - 3q56 + q57 + q58 - 2q59 + q60

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[9, 37]]
Out[2]=	PD[X[1, 4, 2, 5], X[7, 12, 8, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], > X[5, 14, 6, 15], X[13, 6, 14, 7], X[15, 18, 16, 1], X[9, 17, 10, 16], > X[17, 9, 18, 8]]
In[3]:=	GaussCode[Knot[9, 37]]
Out[3]=	GaussCode[-1, 4, -3, 1, -5, 6, -2, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7]
In[4]:=	DTCode[Knot[9, 37]]
Out[4]=	DTCode[4, 10, 14, 12, 16, 2, 6, 18, 8]
In[5]:=	br = BR[Knot[9, 37]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, 1, 4, -3, 2, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[9, 37]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 37]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 37]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, {4, 7}, 2}
In[10]:=	alex = Alexander[Knot[9, 37]][t]
Out[10]=	2 11 2 19 + -- - -- - 11 t + 2 t 2 t t
In[11]:=	Conway[Knot[9, 37]][z]
Out[11]=	2 4 1 - 3 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 37], Knot[11, NonAlternating, 100]}
In[13]:=	{KnotDet[Knot[9, 37]], KnotSignature[Knot[9, 37]]}
Out[13]=	{45, 0}
In[14]:=	Jones[Knot[9, 37]][q]
Out[14]=	-5 3 4 7 8 2 3 4 7 - q + -- - -- + -- - - - 7 q + 5 q - 2 q + q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 37]}
In[16]:=	A2Invariant[Knot[9, 37]][q]
Out[16]=	-16 -14 -12 -10 3 -6 4 6 8 10 12 14 -3 - q + q + q - q + -- + q - 2 q + q + 2 q - q + q + q 8 q
In[17]:=	HOMFLYPT[Knot[9, 37]][a, z]
Out[17]=	2 -4 2 2 2 z 2 2 4 2 4 2 4 -2 + a + 2 a - z - ---- + a z - a z + z + a z 2 a
In[18]:=	Kauffman[Knot[9, 37]][a, z]
Out[18]=	2 2 -4 2 5 z 3 2 2 z z 2 2 -2 + a - 2 a - --- - 7 a z - 2 a z + 12 z - ---- + -- + 14 a z + a 4 2 a a 3 3 4 4 4 2 2 z 6 z 3 3 3 5 3 4 z 3 z > 5 a z - ---- + ---- + 13 a z + 3 a z - 2 a z - 13 z + -- - ---- - 3 a 4 2 a a a 5 5 2 4 4 4 2 z 4 z 5 3 5 5 5 6 > 17 a z - 8 a z + ---- - ---- - 13 a z - 6 a z + a z + 5 z + 3 a a 6 7 3 z 2 6 4 6 3 z 7 3 7 8 2 8 > ---- + 5 a z + 3 a z + ---- + 6 a z + 3 a z + z + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[9, 37]], Vassiliev[3][Knot[9, 37]]}
Out[19]=	{-3, -1}
In[20]:=	Kh[Knot[9, 37]][q, t]
Out[20]=	4 1 2 1 2 2 5 2 3 5 - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 > 4 q t + 3 q t + q t + 4 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[9, 37], 2][q]
Out[21]=	-15 3 9 11 5 26 18 20 46 20 39 59 14 58 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- - -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 48 2 3 4 5 6 7 8 9 10 > -- - 5 q - 44 q + 41 q + q - 29 q + 19 q + 3 q - 11 q + 5 q + q - q 11 12 > 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 937

936 938