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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X18,13,19,14 X6,12,1,11 X20,18,21,17 X16,22,17,21 X4,16,5,15 X14,6,15,5 X12,19,13,20

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 7q-3/2 + 12q-1/2 - 17q1/2 + 19q3/2 - 20q5/2 + 17q7/2 - 13q9/2 + 8q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 - 2q-8 + 3q-6 - 2q-2 + 5 - 2q2 + 2q4 + q6 - q8 + 3q10 - 4q12 + 3q14 + 2q16 - 2q18 + 3q20 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z-1 - 3a-5z - 3a-5z3 + 2a-3z-1 + 6a-3z + 4a-3z3 + 2a-3z5 - 2a-1z-1 - 4a-1z - 2a-1z3 + a-1z5 + az-1 - 2az3 + a3z

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 + 2a-7z - 7a-7z3 + 10a-7z5 - 4a-7z7 + a-6z2 - 6a-6z4 + 13a-6z6 - 6a-6z8 - a-5z-1 + 8a-5z - 24a-5z3 + 25a-5z5 - 2a-5z7 - 4a-5z9 + 8a-4z2 - 31a-4z4 + 39a-4z6 - 14a-4z8 - a-4z10 - 2a-3z-1 + 15a-3z - 35a-3z3 + 25a-3z5 + 4a-3z7 - 8a-3z9 - a-2 + 11a-2z2 - 34a-2z4 + 36a-2z6 - 14a-2z8 - a-2z10 - 2a-1z-1 + 13a-1z - 24a-1z3 + 18a-1z5 - 3a-1z7 - 4a-1z9 + 3z2 - 6z4 + 8z6 - 6z8 - az-1 + 3az - 4az3 + 7az5 - 5az7 - 2a2z2 + 5a2z4 - 3a2z6 - a3z + 2a3z3 - 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                     3   j = 12                   5 1   j = 10                 8 3     j = 8               9 5       j = 6             11 8         j = 4           9 10           j = 2         8 10             j = 0       5 10               j = -2     2 7                 j = -4   1 5                   j = -6   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, 157]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 157]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[18, 13, 19, 14], X[6, 12, 1, 11], X[20, 18, 21, 17], X[16, 22, 17, 21], > X[4, 16, 5, 15], X[14, 6, 15, 5], X[12, 19, 13, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -9, 10, -6}, > {4, -1, 3, -2, 6, -11, 5, -10, 9, -8, 7, -5, 11, -7, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 7 12 3/2 5/2 7/2 -q + ---- - ---- + ------- - 17 Sqrt[q] + 19 q - 20 q + 17 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 13 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 3 2 2 4 6 8 10 12 14 5 + q - -- + -- - -- - 2 q + 2 q + q - q + 3 q - 4 q + 3 q + 8 6 2 q q q 16 18 20 24 > 2 q - 2 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 157]][a, z]
Out[8]=	3 3 3 1 2 2 a z 3 z 6 z 4 z 3 3 z 4 z 2 z -(----) + ---- - --- + - + -- - --- + --- - --- + a z - ---- + ---- - ---- - 5 3 a z z 7 5 3 a 5 3 a a z a z a a a a a 5 5 3 2 z z > 2 a z + ---- + -- 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 157]][a, z]
Out[9]=	-2 1 2 2 a 2 z 8 z 15 z 13 z 3 2 -a - ---- - ---- - --- - - + --- + --- + ---- + ---- + 3 a z - a z + 3 z - 5 3 a z z 7 5 3 a a z a z a a a 2 2 2 2 3 3 3 3 z z 8 z 11 z 2 2 7 z 24 z 35 z 24 z 3 > -- + -- + ---- + ----- - 2 a z - ---- - ----- - ----- - ----- - 4 a z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 3 3 4 2 z 6 z 31 z 34 z 2 4 10 z 25 z > 2 a z - 6 z + ---- - ---- - ----- - ----- + 5 a z + ----- + ----- + 8 6 4 2 7 5 a a a a a a 5 5 6 6 6 6 25 z 18 z 5 3 5 6 z 13 z 39 z 36 z > ----- + ----- + 7 a z - a z + 8 z - -- + ----- + ----- + ----- - 3 a 8 6 4 2 a a a a a 7 7 7 7 8 8 2 6 4 z 2 z 4 z 3 z 7 8 6 z 14 z > 3 a z - ---- - ---- + ---- - ---- - 5 a z - 6 z - ---- - ----- - 7 5 3 a 6 4 a a a a a 8 9 9 9 10 10 14 z 4 z 8 z 4 z z z > ----- - ---- - ---- - ---- - --- - --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 2 5 7 2 10 + 8 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 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 > 9 q t + 10 q t + 11 q t + 8 q t + 9 q t + 5 q t + 8 q t + 10 5 12 5 12 6 14 6 16 7 > 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 L11a157

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