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DrawMorseLink

PD Presentation:

X8192 X14,9,15,10 X6718 X22,15,7,16 X16,6,17,5 X4,22,5,21 X10,4,11,3 X20,18,21,17 X12,20,13,19 X18,12,19,11 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 9q-3/2 + 16q-1/2 - 23q1/2 + 25q3/2 - 27q5/2 + 23q7/2 - 17q9/2 + 11q11/2 - 5q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 2q-6 - 4q-2 + 5 - 3q2 + 5q4 + 4q6 + 5q10 - 6q12 + 2q14 - 2q18 + 3q20 - q22

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-8z4 - a-8z6 - 3a-7z3 + 9a-7z5 - 5a-7z7 - 12a-6z4 + 22a-6z6 - 10a-6z8 + 2a-5z - 7a-5z3 + 5a-5z5 + 12a-5z7 - 9a-5z9 + 2a-4z2 - 29a-4z4 + 47a-4z6 - 16a-4z8 - 3a-4z10 + a-3z-1 + 5a-3z - 12a-3z3 + 26a-3z7 - 17a-3z9 - a-2 + 2a-2z2 - 23a-2z4 + 39a-2z6 - 16a-2z8 - 3a-2z10 + a-1z-1 + 5a-1z - 16a-1z3 + 16a-1z5 + a-1z7 - 8a-1z9 - 2z2 - 2z4 + 11z6 - 10z8 + 2az - 7az3 + 11az5 - 8az7 - 2a2z2 + 5a2z4 - 4a2z6 + a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                     4   j = 12                   7 1   j = 10                 10 4     j = 8               13 7       j = 6             14 10         j = 4           12 14           j = 2         11 13             j = 0       6 13               j = -2     3 10                 j = -4   1 6                   j = -6   3                     j = -8 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, 207]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 207]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 9, 15, 10], X[6, 7, 1, 8], X[22, 15, 7, 16], > X[16, 6, 17, 5], X[4, 22, 5, 21], X[10, 4, 11, 3], X[20, 18, 21, 17], > X[12, 20, 13, 19], X[18, 12, 19, 11], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -6, 5, -3}, > {3, -1, 2, -7, 10, -9, 11, -2, 4, -5, 8, -10, 9, -8, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 9 16 3/2 5/2 7/2 -q + ---- - ---- + ------- - 23 Sqrt[q] + 25 q - 27 q + 23 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 17 q + 11 q - 5 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 4 2 4 6 10 12 14 18 5 + q - -- + -- - -- - 3 q + 5 q + 4 q + 5 q - 6 q + 2 q - 2 q + 8 6 2 q q q 20 22 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 207]][a, z]
Out[8]=	3 3 3 5 5 1 1 z 3 z 2 z z z 4 z 3 z 2 z -(----) + --- - -- + --- - --- + a z + -- + -- - ---- + 2 a z + -- - ---- - 3 a z 5 3 a 5 3 a 5 3 a z a a a a a a 5 7 7 3 z 5 z z > ---- + a z - -- - -- a 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 207]][a, z]
Out[9]=	2 2 -2 1 1 2 z 5 z 5 z 2 2 z 2 z 2 2 -a + ---- + --- + --- + --- + --- + 2 a z - 2 z + ---- + ---- - 2 a z - 3 a z 5 3 a 4 2 a z a a a a 3 3 3 3 4 4 4 3 z 7 z 12 z 16 z 3 3 3 4 z 12 z 29 z > ---- - ---- - ----- - ----- - 7 a z + a z - 2 z + -- - ----- - ----- - 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 6 23 z 2 4 9 z 5 z 16 z 5 3 5 6 z > ----- + 5 a z + ---- + ---- + ----- + 11 a z - a z + 11 z - -- + 2 7 5 a 8 a a a a 6 6 6 7 7 7 7 22 z 47 z 39 z 2 6 5 z 12 z 26 z z 7 > ----- + ----- + ----- - 4 a z - ---- + ----- + ----- + -- - 8 a z - 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 8 10 z 16 z 16 z 9 z 17 z 8 z 3 z 3 z > 10 z - ----- - ----- - ----- - ---- - ----- - ---- - ----- - ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 6 3 6 10 2 13 + 11 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 13 q t + 8 4 6 3 4 3 4 2 2 2 t 2 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 q t + 14 q t + 14 q t + 10 q t + 13 q t + 7 q t + 10 q t + 10 5 12 5 12 6 14 6 16 7 > 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 L11a207

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