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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 7q-7 - 11q-6 + 18q-5 - 20q-4 + 22q-3 - 20q-2 + 15q-1 - 10 + 5q - q2

A2 (sl(3)) Invariant:

q-30 + 3q-28 + 2q-26 + 3q-24 + 8q-22 + 2q-20 + 2q-18 + 6q-16 - 2q-14 + 3q-12 - q-10 - q-8 + 2q-6 - 5q-4 + 5q-2 - 1 - 2q2 + 3q4 - q6

HOMFLY-PT Polynomial:

- z4 + a2 + 4a2z2 + 2a2z4 + a2z6 - 2a4 - 5a4z2 - 4a4z4 + a6z-2 + 5a6 + 6a6z2 - 2a8z-2 - 4a8 + a10z-2

Kauffman Polynomial:

a-1z5 - 5z4 + 5z6 + 5az3 - 15az5 + 10az7 - a2 + 3a2z2 - 4a2z4 - 10a2z6 + 10a2z8 - 2a3z + 24a3z3 - 37a3z5 + 11a3z7 + 5a3z9 - a4 - 2a4z2 + 21a4z4 - 37a4z6 + 18a4z8 + a4z10 - 6a5z + 24a5z3 - 24a5z5 - a5z7 + 8a5z9 + a6z-2 - 3a6 - 2a6z2 + 19a6z4 - 26a6z6 + 11a6z8 + a6z10 - 2a7z-1 + 5a7z3 - 7a7z5 + 3a7z9 + 2a8z-2 - 6a8 + 9a8z2 - 5a8z4 - 3a8z6 + 3a8z8 - 2a9z-1 + 4a9z - 4a9z5 + 2a9z7 + a10z-2 - 4a10 + 6a10z2 - 4a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     4   j = 1                   6 1   j = -1                 9 4     j = -3               12 7       j = -5             10 8         j = -7           10 12           j = -9         8 10             j = -11       4 11               j = -13     3 7                 j = -15   1 6                   j = -17   1                     j = -19 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, 424]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 424]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 7, 17, 8], X[20, 9, 21, 10], > X[18, 22, 19, 21], X[8, 15, 9, 16], X[10, 19, 5, 20], X[22, 14, 11, 13], > X[14, 18, 15, 17], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 4, -7}, > {11, -2, 8, -9, 6, -3, 9, -5, 7, -4, 5, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 7 11 18 20 22 20 15 2 -10 + q - -- + -- - -- + -- - -- + -- - -- + -- + 5 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 3 2 3 8 2 2 6 2 3 -10 -8 -1 + q + --- + --- + --- + --- + --- + --- + --- - --- + --- - q - q + 28 26 24 22 20 18 16 14 12 q q q q q q q q q 2 5 5 2 4 6 > -- - -- + -- - 2 q + 3 q - q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 424]][a, z]
Out[8]=	6 8 10 2 4 6 8 a 2 a a 2 2 4 2 6 2 4 a - 2 a + 5 a - 4 a + -- - ---- + --- + 4 a z - 5 a z + 6 a z - z + 2 2 2 z z z 2 4 4 4 2 6 > 2 a z - 4 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 424]][a, z]
Out[9]=	6 8 10 7 9 2 4 6 8 10 a 2 a a 2 a 2 a 3 -a - a - 3 a - 6 a - 4 a + -- + ---- + --- - ---- - ---- - 2 a z - 2 2 2 z z z z z 5 9 2 2 4 2 6 2 8 2 10 2 > 6 a z + 4 a z + 3 a z - 2 a z - 2 a z + 9 a z + 6 a z + 3 3 3 5 3 7 3 4 2 4 4 4 > 5 a z + 24 a z + 24 a z + 5 a z - 5 z - 4 a z + 21 a z + 5 6 4 8 4 10 4 z 5 3 5 5 5 > 19 a z - 5 a z - 4 a z + -- - 15 a z - 37 a z - 24 a z - a 7 5 9 5 6 2 6 4 6 6 6 8 6 > 7 a z - 4 a z + 5 z - 10 a z - 37 a z - 26 a z - 3 a z + 10 6 7 3 7 5 7 9 7 2 8 4 8 > a z + 10 a z + 11 a z - a z + 2 a z + 10 a z + 18 a z + 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 11 a z + 3 a z + 5 a z + 8 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 1 1 6 3 7 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 11 8 10 10 12 10 8 12 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a424

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