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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X14,6,15,5 X16,8,17,7 X10,4,11,3 X20,12,7,11 X12,15,13,16 X6,14,1,13 X4,19,5,20 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 5q-5/2 - 10q-3/2 + 14q-1/2 - 19q1/2 + 19q3/2 - 18q5/2 + 14q7/2 - 9q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

q-10 - 3q-8 + 2q-6 - q-4 - q-2 + 6 - 2q2 + 5q4 - q6 + q8 + 3q10 - 3q12 + 3q14 - q16 - q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 5a-6z4 - 4a-6z6 + 3a-5z - 9a-5z3 + 13a-5z5 - 8a-5z7 - 2a-4z2 + a-4z4 + 8a-4z6 - 8a-4z8 + a-3z-1 + 4a-3z - 26a-3z3 + 39a-3z5 - 15a-3z7 - 3a-3z9 - a-2 - 2a-2z2 - 4a-2z4 + 25a-2z6 - 17a-2z8 + a-1z-1 - 22a-1z3 + 42a-1z5 - 17a-1z7 - 3a-1z9 - 2z2 + 5z4 + 8z6 - 9z8 - az - 6az3 + 16az5 - 10az7 + 5a2z4 - 5a2z6 - 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 j = 14                     1 j = 12                   3   j = 10                 6 1   j = 8               8 3     j = 6             10 6       j = 4           10 9         j = 2         9 9           j = 0       6 11             j = -2     4 8               j = -4   1 6                 j = -6   4                   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[10, Alternating, 71]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 71]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[14, 6, 15, 5], X[16, 8, 17, 7], > X[10, 4, 11, 3], X[20, 12, 7, 11], X[12, 15, 13, 16], X[6, 14, 1, 13], > X[4, 19, 5, 20], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -9, 3, -8}, > {4, -1, 2, -5, 6, -7, 8, -3, 7, -4, 10, -2, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 5 10 14 3/2 5/2 7/2 -q + ---- - ---- + ------- - 19 Sqrt[q] + 19 q - 18 q + 14 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 3 2 -4 -2 2 4 6 8 10 12 6 + q - -- + -- - q - q - 2 q + 5 q - q + q + 3 q - 3 q + 8 6 q q 14 16 18 20 > 3 q - q - q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 71]][a, z]
Out[8]=	3 3 3 5 5 1 1 z z 2 z z 4 z 2 z 3 2 z 3 z -(----) + --- - -- + -- + --- - a z - -- + ---- - ---- + a z + ---- - ---- + 3 a z 5 3 a 5 3 a 3 a a z a a a a a 7 5 z > a z - -- a
In[9]:=	Kauffman[Link[10, Alternating, 71]][a, z]
Out[9]=	2 2 2 3 3 -2 1 1 3 z 4 z 2 2 z 2 z 2 z z 9 z -a + ---- + --- + --- + --- - a z - 2 z - ---- - ---- - ---- + -- - ---- - 3 a z 5 3 6 4 2 7 5 a z a a a a a a a 3 3 4 4 4 5 5 26 z 22 z 3 4 5 z z 4 z 2 4 z 13 z > ----- - ----- - 6 a z + 5 z + ---- + -- - ---- + 5 a z - -- + ----- + 3 a 6 4 2 7 5 a a a a a a 5 5 6 6 6 39 z 42 z 5 3 5 6 4 z 8 z 25 z 2 6 > ----- + ----- + 16 a z - a z + 8 z - ---- + ---- + ----- - 5 a z - 3 a 6 4 2 a a a a 7 7 7 8 8 9 9 8 z 15 z 17 z 7 8 8 z 17 z 3 z 3 z > ---- - ----- - ----- - 10 a z - 9 z - ---- - ----- - ---- - ---- 5 3 a 4 2 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 4 1 6 4 6 8 2 11 + 9 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 9 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 > 10 q t + 9 q t + 10 q t + 6 q t + 8 q t + 3 q t + 6 q t + 10 5 12 5 14 6 > q t + 3 q t + q t

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