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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X20,15,21,16 X14,6,15,5 X12,4,13,3 X4,14,5,13 X2,19,3,20 X16,7,17,8 X8,9,1,10 X18,12,19,11 X22,18,9,17 X6,22,7,21

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 8q-5/2 - 13q-3/2 + 17q-1/2 - 21q1/2 + 20q3/2 - 18q5/2 + 13q7/2 - 8q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 + 2q-10 - 2q-8 + 3q-6 + 5 - 3q2 + 6q4 - 2q6 + 2q8 + q10 - 3q12 + 2q14 - 2q16 + q18

HOMFLY-PT Polynomial:

- 4a-3z3 - 4a-3z5 - a-3z7 - a-1z-1 + 8a-1z3 + 12a-1z5 + 6a-1z7 + a-1z9 + az-1 - 4az3 - 4az5 - az7

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 6a-6z4 - 4a-6z6 - 3a-5z3 + 11a-5z5 - 7a-5z7 - 4a-4z4 + 12a-4z6 - 8a-4z8 + 5a-3z3 - 11a-3z5 + 12a-3z7 - 7a-3z9 + 6a-2z2 - 23a-2z4 + 20a-2z6 - 4a-2z8 - 3a-2z10 - a-1z-1 + 19a-1z3 - 48a-1z5 + 41a-1z7 - 14a-1z9 + 1 + 8z2 - 28z4 + 24z6 - 3z8 - 3z10 - az-1 + 6az3 - 15az5 + 18az7 - 7az9 + 3a2z2 - 13a2z4 + 19a2z6 - 7a2z8 - 4a3z3 + 10a3z5 - 4a3z7 + 2a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     3   j = 10                   5 1   j = 8                 8 3     j = 6               10 5       j = 4             10 8         j = 2           11 10           j = 0         8 12             j = -2       5 9               j = -4     3 8                 j = -6   1 5                   j = -8   3                     j = -10 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[11, Alternating, 251]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 251]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 15, 21, 16], X[14, 6, 15, 5], X[12, 4, 13, 3], > X[4, 14, 5, 13], X[2, 19, 3, 20], X[16, 7, 17, 8], X[8, 9, 1, 10], > X[18, 12, 19, 11], X[22, 18, 9, 17], X[6, 22, 7, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -6, 4, -5, 3, -11, 7, -8}, > {8, -1, 9, -4, 5, -3, 2, -7, 10, -9, 6, -2, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 8 13 17 3/2 5/2 q - ---- + ---- - ---- + ------- - 21 Sqrt[q] + 20 q - 18 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 13 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 2 3 2 4 6 8 10 12 14 5 - q + --- - -- + -- - 3 q + 6 q - 2 q + 2 q + q - 3 q + 2 q - 10 8 6 q q q 16 18 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 251]][a, z]
Out[8]=	3 3 5 5 7 7 1 a 4 z 8 z 3 4 z 12 z 5 z 6 z 7 -(---) + - - ---- + ---- - 4 a z - ---- + ----- - 4 a z - -- + ---- - a z + a z z 3 a 3 a 3 a a a a 9 z > -- a
In[9]:=	Kauffman[Link[11, Alternating, 251]][a, z]
Out[9]=	2 2 3 3 3 3 1 a 2 z 6 z 2 2 z 3 z 5 z 19 z 3 1 - --- - - + 8 z - -- + ---- + 3 a z + -- - ---- + ---- + ----- + 6 a z - a z z 6 2 7 5 3 a a a a a a 4 4 4 5 5 3 3 4 6 z 4 z 23 z 2 4 4 4 z 11 z > 4 a z - 28 z + ---- - ---- - ----- - 13 a z + 2 a z - -- + ----- - 6 4 2 7 5 a a a a a 5 5 6 6 6 11 z 48 z 5 3 5 6 4 z 12 z 20 z > ----- - ----- - 15 a z + 10 a z + 24 z - ---- + ----- + ----- + 3 a 6 4 2 a a a a 7 7 7 8 2 6 4 6 7 z 12 z 41 z 7 3 7 8 8 z > 19 a z - a z - ---- + ----- + ----- + 18 a z - 4 a z - 3 z - ---- - 5 3 a 4 a a a 8 9 9 10 4 z 2 8 7 z 14 z 9 10 3 z > ---- - 7 a z - ---- - ----- - 7 a z - 3 z - ----- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 8 5 8 12 + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 9 2 4 4 2 6 2 6 3 8 3 > ---- + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + 8 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a251

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