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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X15,20,16,17 X19,16,20,9 X17,12,18,13 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- 3q-9/2 + 4q-7/2 - 9q-5/2 + 7q-3/2 - 9q-1/2 + 7q1/2 - 6q3/2 + 2q5/2 - q7/2

A2 (sl(3)) Invariant:

2q-16 + 6q-14 + 5q-12 + 10q-10 + 12q-8 + 10q-6 + 12q-4 + 6q-2 + 7 + 3q2 + 2q4 + 4q6 + q10 + q12

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-3z-1 - 3a-3z + 3a-3z3 - a-3z5 - a-2 + 3a-2z4 - 2a-2z6 + a-1z-3 - 3a-1z-1 + 4a-1z - 3a-1z3 + 5a-1z5 - 3a-1z7 - 3z-2 + 11 - 18z2 + 21z4 - 7z6 - z8 + 3az-3 - 12az-1 + 20az - 19az3 + 15az5 - 7az7 - 6a2z-2 + 24a2 - 33a2z2 + 24a2z4 - 8a2z6 - a2z8 + 3a3z-3 - 14a3z-1 + 23a3z - 19a3z3 + 9a3z5 - 4a3z7 - 3a4z-2 + 13a4 - 15a4z2 + 6a4z4 - 3a4z6 + a5z-3 - 6a5z-1 + 10a5z - 6a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                 1 j = 6               2 1 j = 4             4     j = 2           3 2     j = 0         6 4       j = -2       5 7         j = -4     4 2           j = -6     5             j = -8 3 4               j = -10 3

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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 99]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 99]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[11, 18, 12, 19], X[15, 20, 16, 17], X[19, 16, 20, 9], X[17, 12, 18, 13], > X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 10}, {9, -1, -3, 4}, {-8, 5, -7, 6}, > {-10, 2, -5, 8, -4, 3, -6, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-3 4 9 7 9 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 6 q + 2 q - q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 5 10 12 10 12 6 2 4 6 10 12 7 + --- + --- + --- + --- + -- + -- + -- + -- + 3 q + 2 q + 4 q + q + q 16 14 12 10 8 6 4 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 99]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 1 5 a 6 a 2 a z 2 z -(----) + --- - ---- + -- - ---- + --- - ---- + ---- - -- + --- + 2 a z - 3 3 3 3 3 z z z 3 a a z z z z a z a 3 3 2 z 3 3 3 5 > 3 a z + ---- - a z + a z - a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 99]][a, z]
Out[9]=	3 5 2 4 -2 2 4 1 3 a 3 a a 3 6 a 3 a 1 11 - a + 24 a + 13 a + ---- + --- + ---- + -- - -- - ---- - ---- + ---- - 3 3 3 3 2 2 2 3 a z z z z z z z a z 3 5 3 12 a 14 a 6 a 3 z 4 z 3 5 > --- - ---- - ----- - ---- - --- + --- + 20 a z + 23 a z + 10 a z - a z z z z 3 a a 3 3 2 2 2 4 2 3 z 3 z 3 3 3 5 3 > 18 z - 33 a z - 15 a z + ---- - ---- - 19 a z - 19 a z - 6 a z + 3 a a 4 5 5 4 3 z 2 4 4 4 z 5 z 5 3 5 6 > 21 z + ---- + 24 a z + 6 a z - -- + ---- + 15 a z + 9 a z - 7 z - 2 3 a a a 6 7 2 z 2 6 4 6 3 z 7 3 7 8 2 8 > ---- - 8 a z - 3 a z - ---- - 7 a z - 4 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 3 3 4 5 4 2 5 2 6 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 4 t + 3 q t + 2 10 4 8 4 8 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 6 3 6 4 8 4 > 2 q t + 4 q t + 2 q t + q t + q t

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