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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,10,17,9 X6,13,7,14 X14,7,15,8 X8,15,1,16 X22,18,9,17 X4,20,5,19 X20,6,21,5 X18,22,19,21

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 2q-13/2 - 4q-11/2 + 6q-9/2 - 8q-7/2 + 9q-5/2 - 9q-3/2 + 7q-1/2 - 7q1/2 + 4q3/2 - 2q5/2 + q7/2

A2 (sl(3)) Invariant:

q-22 + q-18 + q-16 + 2q-12 - q-10 + q-8 + 3q-2 + 1 + 2q2 - q6 - q10

HOMFLY-PT Polynomial:

4a-1z + 4a-1z3 + a-1z5 - 5az - 8az3 - 5az5 - az7 - a3z-1 - 5a3z - 8a3z3 - 5a3z5 - a3z7 + a5z-1 + 4a5z + 4a5z3 + a5z5

Kauffman Polynomial:

8a-2z2 - 12a-2z4 + 6a-2z6 - a-2z8 - 5a-1z + 15a-1z3 - 20a-1z5 + 11a-1z7 - 2a-1z9 + 11z2 - 24z4 + 12z6 + z8 - z10 - 4az + 8az3 - 23az5 + 21az7 - 5az9 + 9a2z2 - 29a2z4 + 24a2z6 - 3a2z8 - a2z10 - a3z-1 + 8a3z - 18a3z3 + 11a3z5 + 5a3z7 - 3a3z9 + a4 + 3a4z2 - 10a4z4 + 14a4z6 - 5a4z8 - a5z-1 + 5a5z - 8a5z3 + 11a5z5 - 5a5z7 - 2a6z2 + 5a6z4 - 4a6z6 - a7z + 2a7z3 - 3a7z5 + a8z2 - 2a8z4 + a9z - a9z3

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 = 8                       1 j = 6                     1   j = 4                   3 1   j = 2                 4 1     j = 0               3 3       j = -2             6 4         j = -4           4 4           j = -6         4 5             j = -8       2 4               j = -10     2 4                 j = -12   1 3                   j = -14   1                     j = -16 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, 324]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 324]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[16, 10, 17, 9], > X[6, 13, 7, 14], X[14, 7, 15, 8], X[8, 15, 1, 16], X[22, 18, 9, 17], > X[4, 20, 5, 19], X[20, 6, 21, 5], X[18, 22, 19, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 10, -5, 6, -7}, > {4, -1, 2, -3, 5, -6, 7, -4, 8, -11, 9, -10, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 4 6 8 9 9 7 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -18 -16 2 -10 -8 3 2 6 10 1 + q + q + q + --- - q + q + -- + 2 q - q - q 12 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 324]][a, z]
Out[8]=	3 5 3 a a 4 z 3 5 4 z 3 3 3 -(--) + -- + --- - 5 a z - 5 a z + 4 a z + ---- - 8 a z - 8 a z + z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 4 a z + -- - 5 a z - 5 a z + a z - a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 324]][a, z]
Out[9]=	3 5 2 4 a a 5 z 3 5 7 9 2 8 z a - -- - -- - --- - 4 a z + 8 a z + 5 a z - a z + a z + 11 z + ---- + z z a 2 a 3 2 2 4 2 6 2 8 2 15 z 3 3 3 5 3 > 9 a z + 3 a z - 2 a z + a z + ----- + 8 a z - 18 a z - 8 a z + a 4 7 3 9 3 4 12 z 2 4 4 4 6 4 8 4 > 2 a z - a z - 24 z - ----- - 29 a z - 10 a z + 5 a z - 2 a z - 2 a 5 6 20 z 5 3 5 5 5 7 5 6 6 z 2 6 > ----- - 23 a z + 11 a z + 11 a z - 3 a z + 12 z + ---- + 24 a z + a 2 a 7 8 4 6 6 6 11 z 7 3 7 5 7 8 z > 14 a z - 4 a z + ----- + 21 a z + 5 a z - 5 a z + z - -- - a 2 a 9 2 8 4 8 2 z 9 3 9 10 2 10 > 3 a z - 5 a z - ---- - 5 a z - 3 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	4 6 1 1 1 3 2 4 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 4 5 4 4 t 2 2 2 2 3 4 3 > ----- + ---- + ---- + 3 t + --- + 3 t + 4 q t + q t + 3 q t + 6 2 6 4 2 q t q t q t q 4 4 6 4 8 5 > q t + q t + q t

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