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DrawMorseLink

PD Presentation:

X8192 X16,6,17,5 X20,10,21,9 X10,22,11,21 X18,16,19,15 X14,20,15,19 X2,11,3,12 X12,3,13,4 X4758 X22,14,7,13 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 8q-3/2 + 13q-1/2 - 19q1/2 + 21q3/2 - 22q5/2 + 19q7/2 - 14q9/2 + 9q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - q-8 + 3q-6 + 2q-4 - q-2 + 6 - 2q2 + 3q4 + q6 - 2q8 + 3q10 - 5q12 + 2q14 - 2q18 + 2q20 - q22

HOMFLY-PT Polynomial:

a-5z + 2a-5z3 + a-5z5 + a-3z-1 + 2a-3z - 2a-3z3 - 3a-3z5 - a-3z7 - 3a-1z-1 - 8a-1z - 8a-1z3 - 4a-1z5 - a-1z7 + 2az-1 + 4az + 3az3 + az5

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 - 5a-7z3 + 9a-7z5 - 4a-7z7 + 2a-6z2 - 10a-6z4 + 16a-6z6 - 7a-6z8 - a-5z + a-5z3 + a-5z5 + 8a-5z7 - 6a-5z9 + a-4 - 2a-4z2 - 11a-4z4 + 26a-4z6 - 10a-4z8 - 2a-4z10 - a-3z-1 + 5a-3z - 9a-3z3 + 2a-3z5 + 15a-3z7 - 11a-3z9 + 3a-2 - 8a-2z2 - 5a-2z4 + 20a-2z6 - 10a-2z8 - 2a-2z10 - 3a-1z-1 + 14a-1z - 27a-1z3 + 21a-1z5 - 3a-1z7 - 5a-1z9 + 3 - 4z2 - 2z4 + 8z6 - 7z8 - 2az-1 + 7az - 10az3 + 10az5 - 6az7 - a2z2 + 4a2z4 - 3a2z6 - a3z + 2a3z3 - 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 r = 7 j = 16                       1 j = 14                     3   j = 12                   6 1   j = 10                 8 3     j = 8               11 6       j = 6             11 8         j = 4           10 11           j = 2         9 11             j = 0       5 11               j = -2     3 8                 j = -4   1 6                   j = -6   2                     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[11, Alternating, 228]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 228]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 6, 17, 5], X[20, 10, 21, 9], X[10, 22, 11, 21], > X[18, 16, 19, 15], X[14, 20, 15, 19], X[2, 11, 3, 12], X[12, 3, 13, 4], > X[4, 7, 5, 8], X[22, 14, 7, 13], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 8, -9, 2, -11}, > {9, -1, 3, -4, 7, -8, 10, -6, 5, -2, 11, -5, 6, -3, 4, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 8 13 3/2 5/2 7/2 -q + ---- - ---- + ------- - 19 Sqrt[q] + 21 q - 22 q + 19 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 14 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 3 2 -2 2 4 6 8 10 12 6 + q - q + -- + -- - q - 2 q + 3 q + q - 2 q + 3 q - 5 q + 6 4 q q 14 18 20 22 > 2 q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 228]][a, z]
Out[8]=	3 3 3 5 1 3 2 a z 2 z 8 z 2 z 2 z 8 z 3 z ---- - --- + --- + -- + --- - --- + 4 a z + ---- - ---- - ---- + 3 a z + -- - 3 a z z 5 3 a 5 3 a 5 a z a a a a a 5 5 7 7 3 z 4 z 5 z z > ---- - ---- + a z - -- - -- 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 228]][a, z]
Out[9]=	2 -4 3 1 3 2 a z 5 z 14 z 3 2 z 3 + a + -- - ---- - --- - --- - -- + --- + ---- + 7 a z - a z - 4 z - -- + 2 3 a z z 5 3 a 8 a a z a a a 2 2 2 3 3 3 3 2 z 2 z 8 z 2 2 5 z z 9 z 27 z 3 3 3 > ---- - ---- - ---- - a z - ---- + -- - ---- - ----- - 10 a z + 2 a z - 6 4 2 7 5 3 a a a a a a a 4 4 4 4 5 5 5 5 4 2 z 10 z 11 z 5 z 2 4 9 z z 2 z 21 z > 2 z + ---- - ----- - ----- - ---- + 4 a z + ---- + -- + ---- + ----- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 6 7 5 3 5 6 z 16 z 26 z 20 z 2 6 4 z > 10 a z - a z + 8 z - -- + ----- + ----- + ----- - 3 a z - ---- + 8 6 4 2 7 a a a a a 7 7 7 8 8 8 9 9 8 z 15 z 3 z 7 8 7 z 10 z 10 z 6 z 11 z > ---- + ----- - ---- - 6 a z - 7 z - ---- - ----- - ----- - ---- - ----- - 5 3 a 6 4 2 5 3 a a a a a a a 9 10 10 5 z 2 z 2 z > ---- - ----- - ----- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 6 3 5 8 2 11 + 9 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 11 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 + 11 q t + 11 q t + 8 q t + 11 q t + 6 q t + 8 q t + 10 5 12 5 12 6 14 6 16 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a228

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