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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,8,15,7 X20,18,21,17 X18,12,19,11 X12,20,13,19 X22,16,5,15 X16,22,17,21 X8,14,9,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 9q-1/2 - 14q1/2 + 16q3/2 - 16q5/2 + 14q7/2 - 11q9/2 + 7q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + q-4 - q-2 + 5 - q2 + 2q4 + q6 - q8 + 2q10 - 4q12 + q14 - 3q18 + 2q20 - q24

HOMFLY-PT Polynomial:

a-7z + a-5z-1 - 2a-5z3 - 2a-3z-1 - 3a-3z - a-3z3 + a-3z5 + a-1z-1 + a-1z + a-1z5 - az-1 - 2az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + a-7z - 5a-7z3 + 8a-7z5 - 3a-7z7 - 3a-6 + 9a-6z2 - 13a-6z4 + 14a-6z6 - 5a-6z8 + a-5z-1 - 2a-5z + a-5z3 - a-5z5 + 7a-5z7 - 4a-5z9 - 7a-4 + 30a-4z2 - 46a-4z4 + 34a-4z6 - 9a-4z8 - a-4z10 + 2a-3z-1 - 10a-3z + 18a-3z3 - 20a-3z5 + 16a-3z7 - 7a-3z9 - 4a-2 + 16a-2z2 - 28a-2z4 + 21a-2z6 - 7a-2z8 - a-2z10 + a-1z-1 - 6a-1z + 10a-1z3 - 7a-1z5 + 3a-1z7 - 3a-1z9 - 3z2 + 5z4 - 3z8 + az-1 - 2az + az3 + 3az5 - 3az7 - a2 + 3a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - 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                     2   j = 12                   5 1   j = 10                 6 2     j = 8               8 5       j = 6             8 6         j = 4           8 8           j = 2         6 8             j = 0       4 9               j = -2     2 5                 j = -4     4                   j = -6 1 2                     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, 26]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 26]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[20, 18, 21, 17], > X[18, 12, 19, 11], X[12, 20, 13, 19], X[22, 16, 5, 15], X[16, 22, 17, 21], > X[8, 14, 9, 13], 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, 7, -8, 4, -5, > 6, -4, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 9 3/2 5/2 7/2 -q + ---- - ---- + ------- - 14 Sqrt[q] + 16 q - 16 q + 14 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 11 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 -4 -2 2 4 6 8 10 12 14 5 + q + q + -- + q - q - q + 2 q + q - q + 2 q - 4 q + q - 6 q 18 20 24 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 26]][a, z]
Out[8]=	3 3 3 1 2 1 a a z 3 z z 3 2 z z 3 ---- - ---- + --- - - + -- + -- - --- + - - 2 a z + a z - ---- - -- - 2 a z + 5 3 a z z z 7 3 a 5 3 a z a z a a a a 5 5 z z > -- + -- 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 26]][a, z]
Out[9]=	3 -3 7 4 2 1 2 1 a a z 2 z 10 z 6 z -- - -- - -- - a + ---- + ---- + --- + - + -- + -- - --- - ---- - --- - 6 4 2 5 3 a z z z 7 5 3 a a a a a z a z a a a 2 2 2 2 3 3 3 3 2 2 z 9 z 30 z 16 z 5 z z 18 z > 2 a z - 3 a z - 3 z - ---- + ---- + ----- + ----- - ---- + -- + ----- + 8 6 4 2 7 5 3 a a a a a a a 3 4 4 4 4 10 z 3 3 3 4 3 z 13 z 46 z 28 z 2 4 > ----- + a z + 3 a z + 5 z + ---- - ----- - ----- - ----- + 3 a z + a 8 6 4 2 a a a a 5 5 5 5 6 6 6 6 8 z z 20 z 7 z 5 3 5 z 14 z 34 z 21 z > ---- - -- - ----- - ---- + 3 a z - a z - -- + ----- + ----- + ----- - 7 5 3 a 8 6 4 2 a a a a a a a 7 7 7 7 8 8 8 2 6 3 z 7 z 16 z 3 z 7 8 5 z 9 z 7 z > 2 a z - ---- + ---- + ----- + ---- - 3 a z - 3 z - ---- - ---- - ---- - 7 5 3 a 6 4 2 a a a a a a 9 9 9 10 10 4 z 7 z 3 z z z > ---- - ---- - ---- - --- - --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 2 4 5 2 4 9 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 8 q t + 8 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 8 q t + 8 q t + 6 q t + 8 q t + 5 q t + 6 q t + 2 q t + 12 5 12 6 14 6 16 7 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a26

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