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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X20,16,21,15 X18,12,19,11 X12,20,13,19 X22,18,5,17 X16,22,17,21 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8, 7, -5, 6, -4, 8, -7}}

Jones Polynomial:

- 2q-7/2 + 4q-5/2 - 8q-3/2 + 10q-1/2 - 12q1/2 + 11q3/2 - 10q5/2 + 7q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-12 + 2q-10 + 4q-6 + q-4 + q-2 + 3 - 2q2 + 3q4 - 2q6 + q8 - 3q12 + q14 - q16

HOMFLY-PT Polynomial:

a-3z-1 + 4a-3z + 3a-3z3 + a-3z5 - 2a-1z-1 - 8a-1z - 10a-1z3 - 5a-1z5 - a-1z7 + az + 2az3 + az5 + a3z-1 + a3z

Kauffman Polynomial:

a-6z2 - a-6z4 + 2a-5z3 - 3a-5z5 + 2a-4 - 6a-4z2 + 8a-4z4 - 6a-4z6 - a-3z-1 + 4a-3z - 9a-3z3 + 11a-3z5 - 7a-3z7 + 5a-2 - 16a-2z2 + 14a-2z4 - a-2z6 - 4a-2z8 - 2a-1z-1 + 8a-1z - 15a-1z3 + 19a-1z5 - 8a-1z7 - a-1z9 + 3 - 8z2 + 7z4 + 5z6 - 5z8 - 2az + 4az3 + 2az5 - az7 - az9 - a2 + a2z2 + 2a2z4 - a2z8 + a3z-1 - 6a3z + 8a3z3 - 3a3z5

Khovanov Homology:

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

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[11, NonAlternating, 63]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 63]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[20, 16, 21, 15], > X[18, 12, 19, 11], X[12, 20, 13, 19], X[22, 18, 5, 17], X[16, 22, 17, 21], > X[13, 8, 14, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8, 7, -5, > 6, -4, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 8 10 3/2 5/2 7/2 ---- + ---- - ---- + ------- - 12 Sqrt[q] + 11 q - 10 q + 7 q - 7/2 5/2 3/2 Sqrt[q] q q q 9/2 11/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 4 -4 -2 2 4 6 8 12 14 16 3 + q + --- + -- + q + q - 2 q + 3 q - 2 q + q - 3 q + q - q 10 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 63]][a, z]
Out[8]=	3 3 3 5 5 1 2 a 4 z 8 z 3 3 z 10 z 3 z 5 z ---- - --- + -- + --- - --- + a z + a z + ---- - ----- + 2 a z + -- - ---- + 3 a z z 3 a 3 a 3 a a z a a a 7 5 z > a z - -- a
In[9]:=	Kauffman[Link[11, NonAlternating, 63]][a, z]
Out[9]=	3 2 2 5 2 1 2 a 4 z 8 z 3 2 z 3 + -- + -- - a - ---- - --- + -- + --- + --- - 2 a z - 6 a z - 8 z + -- - 4 2 3 a z z 3 a 6 a a a z a a 2 2 3 3 3 4 6 z 16 z 2 2 2 z 9 z 15 z 3 3 3 4 z > ---- - ----- + a z + ---- - ---- - ----- + 4 a z + 8 a z + 7 z - -- + 4 2 5 3 a 6 a a a a a 4 4 5 5 5 8 z 14 z 2 4 3 z 11 z 19 z 5 3 5 6 > ---- + ----- + 2 a z - ---- + ----- + ----- + 2 a z - 3 a z + 5 z - 4 2 5 3 a a a a a 6 6 7 7 8 9 6 z z 7 z 8 z 7 8 4 z 2 8 z 9 > ---- - -- - ---- - ---- - a z - 5 z - ---- - a z - -- - a z 4 2 3 a 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 3 1 5 3 5 5 2 7 + 6 q + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 5 q t + 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 12 5 > 6 q t + 5 q t + 5 q t + 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n63

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