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Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X16,7,17,8 X8394 X2,18,3,17 X14,6,15,5 X6,12,7,11 X18,10,19,9 X20,15,11,16 X10,13,1,14 X4,19,5,20

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 9q-9/2 + 15q-7/2 - 19q-5/2 + 20q-3/2 - 20q-1/2 + 15q1/2 - 11q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 - q-18 - q-16 + 3q-14 - 4q-12 + 2q-10 - 2q-6 + 5q-4 - 2q-2 + 7 + 3q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - a-1z3 - a-1z5 + az-1 + az + 3az3 + 3az5 + az7 - 2a3z - 4a3z3 - 2a3z5 + a5z + a5z3

Kauffman Polynomial:

- a-3z5 + 4a-2z4 - 5a-2z6 - a-1z-1 - 6a-1z3 + 17a-1z5 - 11a-1z7 + 1 + z2 - 2z4 + 14z6 - 11z8 - az-1 + 2az - 19az3 + 39az5 - 15az7 - 4az9 + 2a2z2 - 14a2z4 + 34a2z6 - 20a2z8 + 4a3z - 22a3z3 + 34a3z5 - 12a3z7 - 4a3z9 - a4z2 - 3a4z4 + 11a4z6 - 9a4z8 + 2a5z - 8a5z3 + 12a5z5 - 8a5z7 - 2a6z2 + 5a6z4 - 4a6z6 + a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                   4   j = 4                 7 1   j = 2               8 4     j = 0             12 7       j = -2           10 10         j = -4         9 10           j = -6       6 10             j = -8     3 9               j = -10   1 6                 j = -12   3                   j = -14 1

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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 111]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 111]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 7, 17, 8], X[8, 3, 9, 4], X[2, 18, 3, 17], > X[14, 6, 15, 5], X[6, 12, 7, 11], X[18, 10, 19, 9], X[20, 15, 11, 16], > X[10, 13, 1, 14], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10, 5, -6, 2, -3, 7, -9}, > {6, -1, 9, -5, 8, -2, 4, -7, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 9 15 19 20 20 -q + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 > 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 3 4 2 2 5 2 6 8 10 7 + q - q - q + --- - --- + --- - -- + -- - -- + 3 q - 3 q + q 14 12 10 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 111]][a, z]
Out[8]=	3 5 1 a 3 5 z 3 3 3 5 3 z -(---) + - + a z - 2 a z + a z - -- + 3 a z - 4 a z + a z - -- + a z z a a 5 3 5 7 > 3 a z - 2 a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 111]][a, z]
Out[9]=	3 1 a 3 5 2 2 2 4 2 6 2 6 z 1 - --- - - + 2 a z + 4 a z + 2 a z + z + 2 a z - a z - 2 a z - ---- - a z z a 4 3 3 3 5 3 7 3 4 4 z 2 4 4 4 > 19 a z - 22 a z - 8 a z + a z - 2 z + ---- - 14 a z - 3 a z + 2 a 5 5 6 4 z 17 z 5 3 5 5 5 7 5 6 > 5 a z - -- + ----- + 39 a z + 34 a z + 12 a z - a z + 14 z - 3 a a 6 7 5 z 2 6 4 6 6 6 11 z 7 3 7 > ---- + 34 a z + 11 a z - 4 a z - ----- - 15 a z - 12 a z - 2 a a 5 7 8 2 8 4 8 9 3 9 > 8 a z - 11 z - 20 a z - 9 a z - 4 a z - 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 6 3 9 6 10 12 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 9 10 10 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 7 t + 8 q t + 4 q t + 7 q t + q t + 4 q t + 4 2 4 2 q t q t q t 8 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a111

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