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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X20,11,21,12 X18,8,19,7 X16,10,17,9 X8,18,9,17 X22,19,13,20 X10,14,11,13 X12,21,5,22 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

- q-5 + 3q-4 - 7q-3 + 13q-2 - 16q-1 + 19 - 18q + 17q2 - 11q3 + 7q4 - 3q5 + q6

A2 (sl(3)) Invariant:

- q-16 + q-12 - 3q-10 + 3q-8 + 2q-6 - q-4 + 5q-2 - 1 + 5q2 + 3q4 + 3q6 + 7q8 - q10 + 3q12 + 2q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4z-2 + 2a-4 + 2a-4z2 + a-4z4 - 2a-2z-2 - 5a-2 - 6a-2z2 - 3a-2z4 - a-2z6 + z-2 + 2 + z2 - z4 - z6 + 2a2 + 3a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial:

2a-6z2 - 3a-6z4 + a-6z6 + 5a-5z3 - 8a-5z5 + 3a-5z7 - a-4z-2 + 6a-4 - 13a-4z2 + 16a-4z4 - 14a-4z6 + 5a-4z8 + 2a-3z-1 - 5a-3z + 6a-3z3 - 3a-3z5 - 5a-3z7 + 4a-3z9 - 2a-2z-2 + 13a-2 - 39a-2z2 + 55a-2z4 - 41a-2z6 + 12a-2z8 + a-2z10 + 2a-1z-1 - 8a-1z + 8a-1z3 - 11a-1z7 + 8a-1z9 - z-2 + 9 - 28z2 + 44z4 - 36z6 + 13z8 + z10 - 3az + 10az3 - 12az5 + 2az7 + 4az9 - a2z2 + 3a2z4 - 7a2z6 + 6a2z8 + a3z + a3z3 - 6a3z5 + 5a3z7 - a4 + 3a4z2 - 5a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     2   j = 9                   5 1   j = 7                 7 3     j = 5               10 4       j = 3             8 7         j = 1           11 10           j = -1         8 11             j = -3       5 8               j = -5     2 8                 j = -7   1 5                   j = -9   2                     j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 456]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 456]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[20, 11, 21, 12], X[18, 8, 19, 7], > X[16, 10, 17, 9], X[8, 18, 9, 17], X[22, 19, 13, 20], X[10, 14, 11, 13], > X[12, 21, 5, 22], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -6, 5, -8, 3, -9}, > {8, -2, 11, -5, 6, -4, 7, -3, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 3 7 13 16 2 3 4 5 6 19 - q + -- - -- + -- - -- - 18 q + 17 q - 11 q + 7 q - 3 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -12 3 3 2 -4 5 2 4 6 8 10 -1 - q + q - --- + -- + -- - q + -- + 5 q + 3 q + 3 q + 7 q - q + 10 8 6 2 q q q q 12 14 16 18 > 3 q + 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 456]][a, z]
Out[8]=	2 2 2 5 2 4 -2 1 2 2 2 z 6 z 2 2 2 + -- - -- + 2 a - a + z + ----- - ----- + z + ---- - ---- + 3 a z - 4 2 4 2 2 2 4 2 a a a z a z a a 4 4 6 4 2 4 z 3 z 2 4 6 z > a z - z + -- - ---- + 2 a z - z - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 456]][a, z]
Out[9]=	6 13 4 -2 1 2 2 2 5 z 8 z 9 + -- + -- - a - z - ----- - ----- + ---- + --- - --- - --- - 3 a z + 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 3 3 5 2 2 z 13 z 39 z 2 2 4 2 5 z > a z + a z - 28 z + ---- - ----- - ----- - a z + 3 a z + ---- + 6 4 2 5 a a a a 3 3 4 4 4 6 z 8 z 3 3 3 5 3 4 3 z 16 z 55 z > ---- + ---- + 10 a z + a z - 2 a z + 44 z - ---- + ----- + ----- + 3 a 6 4 2 a a a a 5 5 6 2 4 4 4 8 z 3 z 5 3 5 5 5 6 z > 3 a z - 5 a z - ---- - ---- - 12 a z - 6 a z + a z - 36 z + -- - 5 3 6 a a a 6 6 7 7 7 14 z 41 z 2 6 4 6 3 z 5 z 11 z 7 > ----- - ----- - 7 a z + 3 a z + ---- - ---- - ----- + 2 a z + 4 2 5 3 a a a a a 8 8 9 9 10 3 7 8 5 z 12 z 2 8 4 z 8 z 9 10 z > 5 a z + 13 z + ---- + ----- + 6 a z + ---- + ---- + 4 a z + z + --- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 2 1 5 2 8 5 8 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 8 3 3 2 5 2 5 3 7 3 7 4 > --- + 10 q t + 8 q t + 7 q t + 10 q t + 4 q t + 7 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 5 q t + q t + 2 q t + q t

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