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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-28 + 3q-24 + 4q-22 + 6q-20 + 7q-18 + 5q-16 + 5q-14 + q-10 - q-8 - q-6 - q-4 - q-2 - q2

HOMFLY-PT Polynomial:

- a2z-2 - 4a2 - 7a2z2 - 5a2z4 - a2z6 + 4a4z-2 + 14a4 + 23a4z2 + 18a4z4 + 7a4z6 + a4z8 - 5a6z-2 - 12a6 - 12a6z2 - 6a6z4 - a6z6 + 2a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

az-1 - 4az + 7az3 - 5az5 + az7 - a2z-2 + 3a2 - 8a2z2 + 14a2z4 - 10a2z6 + 2a2z8 + 5a3z-1 - 19a3z + 29a3z3 - 14a3z5 - a3z7 + a3z9 - 4a4z-2 + 15a4 - 32a4z2 + 42a4z4 - 26a4z6 + 5a4z8 + 9a5z-1 - 33a5z + 41a5z3 - 19a5z5 + a5z9 - 5a6z-2 + 20a6 - 33a6z2 + 31a6z4 - 16a6z6 + 3a6z8 + 5a7z-1 - 18a7z + 20a7z3 - 10a7z5 + 2a7z7 - 2a8z-2 + 8a8 - 8a8z2 + 3a8z4 + a9z3 - a10 + a10z2

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 3                   1 j = 1                 1   j = -1               2 1   j = -3           1 3 1     j = -5           3 3       j = -7       1 4 2         j = -9     1 2 3           j = -11     4 3             j = -13 1   2               j = -15 2 2                 j = -17 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, 274]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 274]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[17, 11, 18, 22], X[11, 21, 12, 20], > X[21, 17, 22, 16], X[4, 15, 1, 16], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, > {-8, 2, -4, -5, 10, 9, -7, 6, -11, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-8 -7 4 3 5 4 4 3 2 + q - q + -- - -- + -- - -- + -- - - - q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 4 6 7 5 5 -10 -8 -6 -4 -2 2 q + --- + --- + --- + --- + --- + --- + q - q - q - q - q - q 24 22 20 18 16 14 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 274]][a, z]
Out[8]=	2 4 6 8 2 4 6 8 a 4 a 5 a 2 a 2 2 4 2 -4 a + 14 a - 12 a + 2 a - -- + ---- - ---- + ---- - 7 a z + 23 a z - 2 2 2 2 z z z z 6 2 8 2 2 4 4 4 6 4 2 6 4 6 6 6 > 12 a z + a z - 5 a z + 18 a z - 6 a z - a z + 7 a z - a z + 4 8 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 274]][a, z]
Out[9]=	2 4 6 8 3 5 2 4 6 8 10 a 4 a 5 a 2 a a 5 a 9 a 3 a + 15 a + 20 a + 8 a - a - -- - ---- - ---- - ---- + - + ---- + ---- + 2 2 2 2 z z z z z z z 7 5 a 3 5 7 2 2 4 2 > ---- - 4 a z - 19 a z - 33 a z - 18 a z - 8 a z - 32 a z - z 6 2 8 2 10 2 3 3 3 5 3 7 3 > 33 a z - 8 a z + a z + 7 a z + 29 a z + 41 a z + 20 a z + 9 3 2 4 4 4 6 4 8 4 5 3 5 > a z + 14 a z + 42 a z + 31 a z + 3 a z - 5 a z - 14 a z - 5 5 7 5 2 6 4 6 6 6 7 3 7 > 19 a z - 10 a z - 10 a z - 26 a z - 16 a z + a z - a z + 7 7 2 8 4 8 6 8 3 9 5 9 > 2 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 2 1 2 2 4 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 5 3 17 6 15 6 13 6 15 5 13 4 11 4 9 4 q q q t q t q t q t q t q t q t 3 2 1 3 4 2 3 1 t 2 t > ------ + ----- + ----- + ----- + ----- + ---- + ---- + ---- + -- + --- + 11 3 9 3 7 3 9 2 7 2 7 5 3 3 q q t q t q t q t q t q t q t q t q 2 t 2 3 3 > -- + q t + q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n274

L11n273 L11n275