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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X13,21,14,20 X19,11,20,22 X10,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X21,15,22,14 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 5q-7 - 6q-6 + 9q-5 - 8q-4 + 9q-3 - 6q-2 + 4q-1 - 2

A2 (sl(3)) Invariant:

q-28 + 2q-26 + 2q-24 + 5q-22 + 3q-20 + 5q-18 + 6q-16 + 3q-14 + 5q-12 + q-8 - q-6 - 3q-4 - 2

HOMFLY-PT Polynomial:

- a2z-2 - 5a2 - 6a2z2 - 2a2z4 + 4a4z-2 + 17a4 + 22a4z2 + 11a4z4 + 2a4z6 - 5a6z-2 - 15a6 - 12a6z2 - 3a6z4 + 2a8z-2 + 3a8 + a8z2

Kauffman Polynomial:

az-1 - 5az + 3az3 - a2z-2 + 4a2 - 5a2z2 + 2a2z4 + a2z6 + 5a3z-1 - 21a3z + 30a3z3 - 15a3z5 + 4a3z7 - 4a4z-2 + 17a4 - 32a4z2 + 34a4z4 - 16a4z6 + 4a4z8 + 9a5z-1 - 33a5z + 43a5z3 - 23a5z5 + 4a5z7 + a5z9 - 5a6z-2 + 20a6 - 36a6z2 + 36a6z4 - 22a6z6 + 6a6z8 + 5a7z-1 - 16a7z + 19a7z3 - 14a7z5 + 2a7z7 + a7z9 - 2a8z-2 + 6a8 - 4a8z2 - 4a8z6 + 2a8z8 + a9z + 3a9z3 - 6a9z5 + 2a9z7 - 2a10 + 5a10z2 - 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 j = 1                   2 j = -1                 2   j = -3               5 3   j = -5             4 1     j = -7           4 5       j = -9         5 4         j = -11       2 5           j = -13     3 4             j = -15   1 4               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, NonAlternating, 305]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 305]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 21, 14, 20], X[19, 11, 20, 22], > X[10, 15, 5, 16], X[8, 17, 9, 18], X[16, 7, 17, 8], X[18, 9, 19, 10], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 7, -6, 8, -5}, > {11, -2, -3, 9, 5, -7, 6, -8, -4, 3, -9, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 5 6 9 8 9 6 4 -2 + q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 2 5 3 5 6 3 5 -8 -6 3 -2 + q + --- + --- + --- + --- + --- + --- + --- + --- + q - q - -- 26 24 22 20 18 16 14 12 4 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 305]][a, z]
Out[8]=	2 4 6 8 2 4 6 8 a 4 a 5 a 2 a 2 2 4 2 -5 a + 17 a - 15 a + 3 a - -- + ---- - ---- + ---- - 6 a z + 22 a z - 2 2 2 2 z z z z 6 2 8 2 2 4 4 4 6 4 4 6 > 12 a z + a z - 2 a z + 11 a z - 3 a z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 305]][a, z]
Out[9]=	2 4 6 8 3 2 4 6 8 10 a 4 a 5 a 2 a a 5 a 4 a + 17 a + 20 a + 6 a - 2 a - -- - ---- - ---- - ---- + - + ---- + 2 2 2 2 z z z z z z 5 7 9 a 5 a 3 5 7 9 2 2 > ---- + ---- - 5 a z - 21 a z - 33 a z - 16 a z + a z - 5 a z - z z 4 2 6 2 8 2 10 2 3 3 3 5 3 > 32 a z - 36 a z - 4 a z + 5 a z + 3 a z + 30 a z + 43 a z + 7 3 9 3 2 4 4 4 6 4 10 4 3 5 > 19 a z + 3 a z + 2 a z + 34 a z + 36 a z - 4 a z - 15 a z - 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 23 a z - 14 a z - 6 a z + a z - 16 a z - 22 a z - 4 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > a z + 4 a z + 4 a z + 2 a z + 2 a z + 4 a z + 6 a z + 8 8 5 9 7 9 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 1 1 1 4 3 4 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 5 5 4 4 5 4 1 5 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 2 q t 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n305

L11n304 L11n306