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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 5q-7/2 + 11q-5/2 - 18q-3/2 + 24q-1/2 - 29q1/2 + 28q3/2 - 25q5/2 + 18q7/2 - 11q9/2 + 5q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 + 3q-10 - 3q-8 + 4q-6 - q-4 - q-2 + 6 - 5q2 + 8q4 - 3q6 + 3q8 + 2q10 - 4q12 + 3q14 - 3q16 + q18

HOMFLY-PT Polynomial:

2a-3z - a-3z3 - 3a-3z5 - a-3z7 - a-1z-1 - 4a-1z + 7a-1z5 + 5a-1z7 + a-1z9 + az-1 + 2az - az3 - 3az5 - az7

Kauffman Polynomial:

- a-7z5 + 4a-6z4 - 5a-6z6 - 4a-5z3 + 14a-5z5 - 11a-5z7 - a-4z2 - 4a-4z4 + 18a-4z6 - 14a-4z8 + 4a-3z - 13a-3z3 + 14a-3z5 + 6a-3z7 - 11a-3z9 + 3a-2z2 - 30a-2z4 + 48a-2z6 - 18a-2z8 - 4a-2z10 - a-1z-1 + 8a-1z - 15a-1z3 - 8a-1z5 + 37a-1z7 - 21a-1z9 + 1 + 5z2 - 34z4 + 47z6 - 14z8 - 4z10 - az-1 + 4az - 9az3 + 2az5 + 15az7 - 10az9 + a2z2 - 11a2z4 + 21a2z6 - 10a2z8 - 3a3z3 + 9a3z5 - 5a3z7 + a4z4 - 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                     4   j = 10                   7 1   j = 8                 11 4     j = 6               14 7       j = 4             14 11         j = 2           15 14           j = 0         11 16             j = -2       7 13               j = -4     4 11                 j = -6   1 7                   j = -8   4                     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, 249]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 249]]
Out[4]=	PD[X[10, 1, 11, 2], X[16, 8, 17, 7], X[18, 12, 19, 11], X[2, 19, 3, 20], > X[12, 4, 13, 3], X[20, 13, 21, 14], X[14, 5, 15, 6], X[6, 9, 7, 10], > X[22, 16, 9, 15], X[8, 18, 1, 17], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 5, -11, 7, -8, 2, -10}, > {8, -1, 3, -5, 6, -7, 9, -2, 10, -3, 4, -6, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 5 11 18 24 3/2 5/2 q - ---- + ---- - ---- + ------- - 29 Sqrt[q] + 28 q - 25 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 18 q - 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 3 3 4 -4 -2 2 4 6 8 10 6 - q + --- - -- + -- - q - q - 5 q + 8 q - 3 q + 3 q + 2 q - 10 8 6 q q q 12 14 16 18 > 4 q + 3 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 249]][a, z]
Out[8]=	3 5 5 7 7 1 a 2 z 4 z z 3 3 z 7 z 5 z 5 z -(---) + - + --- - --- + 2 a z - -- - a z - ---- + ---- - 3 a z - -- + ---- - a z z 3 a 3 3 a 3 a a a a a 9 7 z > a z + -- a
In[9]:=	Kauffman[Link[11, Alternating, 249]][a, z]
Out[9]=	2 2 3 3 1 a 4 z 8 z 2 z 3 z 2 2 4 z 13 z 1 - --- - - + --- + --- + 4 a z + 5 z - -- + ---- + a z - ---- - ----- - a z z 3 a 4 2 5 3 a a a a a 3 4 4 4 15 z 3 3 3 4 4 z 4 z 30 z 2 4 4 4 > ----- - 9 a z - 3 a z - 34 z + ---- - ---- - ----- - 11 a z + a z - a 6 4 2 a a a 5 5 5 5 6 6 z 14 z 14 z 8 z 5 3 5 6 5 z 18 z > -- + ----- + ----- - ---- + 2 a z + 9 a z + 47 z - ---- + ----- + 7 5 3 a 6 4 a a a a a 6 7 7 7 48 z 2 6 4 6 11 z 6 z 37 z 7 3 7 > ----- + 21 a z - a z - ----- + ---- + ----- + 15 a z - 5 a z - 2 5 3 a a a a 8 8 9 9 10 8 14 z 18 z 2 8 11 z 21 z 9 10 4 z > 14 z - ----- - ----- - 10 a z - ----- - ----- - 10 a z - 4 z - ----- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 4 1 7 4 11 7 11 16 + 15 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 13 2 4 4 2 6 2 6 3 8 3 > ---- + 14 q t + 14 q t + 11 q t + 14 q t + 7 q t + 11 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a249

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