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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X20,9,21,10 X8493 X18,22,19,21 X11,14,12,15 X5,13,6,12 X13,5,14,22 X10,19,11,20 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

- 2q-9/2 + 5q-7/2 - 9q-5/2 + 11q-3/2 - 12q-1/2 + 12q1/2 - 10q3/2 + 6q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-16 + 3q-14 - q-12 + q-10 + 2q-8 - 3q-6 + q-4 - 3q-2 + q2 + 5q6 + q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-3z-1 + a-3z3 + a-1z-1 - a-1z3 - a-1z5 + az-1 + 2az - az5 - 2a3z-1 - 2a3z + a3z3 + a5z-1

Kauffman Polynomial:

2a-4z4 - a-4z6 + a-3z-1 - a-3z - 8a-3z3 + 12a-3z5 - 4a-3z7 - a-2 + 4a-2z2 - 10a-2z4 + 14a-2z6 - 5a-2z8 + a-1z-1 + a-1z - 19a-1z3 + 24a-1z5 - 4a-1z7 - 2a-1z9 - 2 + 14z2 - 27z4 + 28z6 - 10z8 - az-1 + 10az - 22az3 + 19az5 - 4az7 - 2az9 - 3a2 + 13a2z2 - 19a2z4 + 12a2z6 - 5a2z8 - 2a3z-1 + 11a3z - 14a3z3 + 7a3z5 - 4a3z7 - a4 + 3a4z2 - 4a4z4 - a4z6 - a5z-1 + 3a5z - 3a5z3

Khovanov Homology:

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

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, NonAlternating, 25]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 25]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[20, 9, 21, 10], > X[8, 4, 9, 3], X[18, 22, 19, 21], X[11, 14, 12, 15], X[5, 13, 6, 12], > X[13, 5, 14, 22], X[10, 19, 11, 20], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-8, -1, 2, -5, 4, -10, -7, 8, -9, 7, 11, -2, 3, -6, > 10, -4, 6, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 5 9 11 12 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 10 q + 6 q - 4 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 3 -12 -10 2 3 -4 3 2 6 10 12 14 q + --- - q + q + -- - -- + q - -- + q + 5 q + q + 2 q - q 14 8 6 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 25]][a, z]
Out[8]=	3 5 3 3 5 1 1 a 2 a a 3 z z 3 3 z 5 -(----) + --- + - - ---- + -- + 2 a z - 2 a z + -- - -- + a z - -- - a z 3 a z z z z 3 a a a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 25]][a, z]
Out[9]=	3 5 -2 2 4 1 1 a 2 a a z z 3 -2 - a - 3 a - a + ---- + --- - - - ---- - -- - -- + - + 10 a z + 11 a z + 3 a z z z z 3 a a z a 2 3 3 5 2 4 z 2 2 4 2 8 z 19 z 3 > 3 a z + 14 z + ---- + 13 a z + 3 a z - ---- - ----- - 22 a z - 2 3 a a a 4 4 5 3 3 5 3 4 2 z 10 z 2 4 4 4 12 z > 14 a z - 3 a z - 27 z + ---- - ----- - 19 a z - 4 a z + ----- + 4 2 3 a a a 5 6 6 7 24 z 5 3 5 6 z 14 z 2 6 4 6 4 z > ----- + 19 a z + 7 a z + 28 z - -- + ----- + 12 a z - a z - ---- - a 4 2 3 a a a 7 8 9 4 z 7 3 7 8 5 z 2 8 2 z 9 > ---- - 4 a z - 4 a z - 10 z - ---- - 5 a z - ---- - 2 a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 2 3 2 6 3 5 6 2 7 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 7 t + 5 q t + 2 10 4 8 3 6 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 4 3 6 3 6 4 8 4 10 5 > 3 q t + 7 q t + 3 q t + 3 q t + q t + 3 q t + q t

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