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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 7q-7/2 + 10q-5/2 - 14q-3/2 + 15q-1/2 - 16q1/2 + 14q3/2 - 10q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + q-16 - q-14 + 3q-12 - q-8 + 4q-6 - q-4 + 3q-2 + 2 + 2q4 - 4q6 + q8 - 2q12 + 2q14 - q18

HOMFLY-PT Polynomial:

a-5z + a-3z-1 - a-3z - 2a-3z3 - 3a-1z-1 - 4a-1z - a-1z3 + a-1z5 + 3az-1 + 6az + 5az3 + 2az5 - 2a3z-1 - 5a3z - 3a3z3 + a5z-1 + a5z

Kauffman Polynomial:

a-6z2 - a-6z4 - a-5z + 3a-5z3 - 3a-5z5 + a-4 - 3a-4z2 + 5a-4z4 - 5a-4z6 - a-3z-1 + a-3z - a-3z3 + 5a-3z5 - 6a-3z7 + 2a-2 - 8a-2z2 + 5a-2z4 + 5a-2z6 - 6a-2z8 - 3a-1z-1 + 13a-1z - 23a-1z3 + 19a-1z5 - a-1z7 - 4a-1z9 + 6z2 - 27z4 + 35z6 - 11z8 - z10 - 3az-1 + 18az - 35az3 + 18az5 + 11az7 - 7az9 - 2a2 + 14a2z2 - 38a2z4 + 36a2z6 - 8a2z8 - a2z10 - 2a3z-1 + 11a3z - 22a3z3 + 11a3z5 + 5a3z7 - 3a3z9 + 4a4z2 - 12a4z4 + 11a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 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                   4 1   j = 6                 6 2     j = 4               8 4       j = 2             8 6         j = 0           8 9           j = -2         6 7             j = -4       4 8               j = -6     3 6                 j = -8   1 5                   j = -10   2                     j = -12 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, 35]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 35]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 8, 17, 7], X[18, 13, 19, 14], > X[14, 17, 15, 18], X[22, 20, 5, 19], X[20, 12, 21, 11], X[12, 22, 13, 21], > X[8, 16, 9, 15], 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, 7, -8, 4, -5, 9, -3, 5, -4, > 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 7 10 14 15 3/2 -q + ---- - ---- + ---- - ---- + ------- - 16 Sqrt[q] + 14 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 10 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 3 -8 4 -4 3 4 6 8 12 2 + q + q - q + --- - q + -- - q + -- + 2 q - 4 q + q - 2 q + 12 6 2 q q q 14 18 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 35]][a, z]
Out[8]=	3 5 3 1 3 3 a 2 a a z z 4 z 3 5 2 z ---- - --- + --- - ---- + -- + -- - -- - --- + 6 a z - 5 a z + a z - ---- - 3 a z z z z 5 3 a 3 a z a a a 3 5 z 3 3 3 z 5 > -- + 5 a z - 3 a z + -- + 2 a z a a
In[9]:=	Kauffman[Link[11, Alternating, 35]][a, z]
Out[9]=	3 5 -4 2 2 1 3 3 a 2 a a z z 13 z a + -- - 2 a - ---- - --- - --- - ---- - -- - -- + -- + ---- + 18 a z + 2 3 a z z z z 5 3 a a a z a a 2 2 2 3 3 5 2 z 3 z 8 z 2 2 4 2 3 z > 11 a z + 4 a z + 6 z + -- - ---- - ---- + 14 a z + 4 a z + ---- - 6 4 2 5 a a a a 3 3 4 4 4 z 23 z 3 3 3 5 3 4 z 5 z 5 z > -- - ----- - 35 a z - 22 a z - 6 a z - 27 z - -- + ---- + ---- - 3 a 6 4 2 a a a a 5 5 5 2 4 4 4 3 z 5 z 19 z 5 3 5 5 5 > 38 a z - 12 a z - ---- + ---- + ----- + 18 a z + 11 a z + 4 a z + 5 3 a a a 6 6 7 7 6 5 z 5 z 2 6 4 6 6 z z 7 3 7 > 35 z - ---- + ---- + 36 a z + 11 a z - ---- - -- + 11 a z + 5 a z - 4 2 3 a a a a 8 9 5 7 8 6 z 2 8 4 8 4 z 9 3 9 10 > a z - 11 z - ---- - 8 a z - 3 a z - ---- - 7 a z - 3 a z - z - 2 a a 2 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 3 6 4 8 9 + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 6 8 7 2 4 4 2 6 2 6 3 > ----- + - + ---- + 6 q t + 8 q t + 4 q t + 6 q t + 2 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a35

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