© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a243 K11a245

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X18,6,19,5 X22,8,1,7 X16,10,17,9 X20,12,21,11 X2,14,3,13 X8,16,9,15 X12,18,13,17 X6,20,7,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 18 22 16 20 2 8 12 6 10

Alexander Polynomial:

5t-3 - 17t-2 + 32t-1 - 39 + 32t - 17t2 + 5t3

Conway Polynomial:

1 + 9z2 + 13z4 + 5z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{147, 6}

Jones Polynomial:

q3 - 3q4 + 9q5 - 14q6 + 20q7 - 24q8 + 24q9 - 21q10 + 16q11 - 10q12 + 4q13 - q14

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q10 - 2q12 + 4q14 - q16 + q18 + 5q20 - 4q22 + 4q24 - 3q26 + 2q30 - 4q32 + 3q34 - 3q36 - q38 + 2q40 - q42

HOMFLY-PT Polynomial:

- a-12z2 - a-12z4 - 3a-10 - 5a-10z2 + a-10z6 + 3a-8 + 12a-8z2 + 11a-8z4 + 3a-8z6 + a-6 + 3a-6z2 + 3a-6z4 + a-6z6

Kauffman Polynomial:

- a-17z3 + a-17z5 + a-16z2 - 4a-16z4 + 4a-16z6 - 4a-15z + 10a-15z3 - 14a-15z5 + 9a-15z7 - 3a-14z2 + 9a-14z4 - 16a-14z6 + 11a-14z8 - 5a-13z + 20a-13z3 - 24a-13z5 + 3a-13z7 + 7a-13z9 + 17a-12z4 - 36a-12z6 + 17a-12z8 + 2a-12z10 - 7a-11z + 25a-11z3 - 20a-11z5 - 10a-11z7 + 12a-11z9 + 3a-10 - 10a-10z2 + 22a-10z4 - 31a-10z6 + 12a-10z8 + 2a-10z10 - 6a-9z + 19a-9z3 - 17a-9z5 - a-9z7 + 5a-9z9 + 3a-8 - 11a-8z2 + 15a-8z4 - 14a-8z6 + 6a-8z8 + 3a-7z3 - 6a-7z5 + 3a-7z7 - a-6 + 3a-6z2 - 3a-6z4 + a-6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{9, 26}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=6 is the signature of 11244. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                     3   j = 25                   7 1   j = 23                 9 3     j = 21               12 7       j = 19             12 9         j = 17           12 12           j = 15         8 12             j = 13       6 12               j = 11     3 8                 j = 9     6                   j = 7 1 3                     j = 5 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 244]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 244]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[22, 8, 1, 7], > X[16, 10, 17, 9], X[20, 12, 21, 11], X[2, 14, 3, 13], X[8, 16, 9, 15], > X[12, 18, 13, 17], X[6, 20, 7, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 244]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -10, 4, -8, 5, -11, 6, -9, 7, -2, 8, -5, 9, -3, 10, > -6, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 244]]
Out[5]=	DTCode[4, 14, 18, 22, 16, 20, 2, 8, 12, 6, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 244]][t]
Out[6]=	5 17 32 2 3 -39 + -- - -- + -- + 32 t - 17 t + 5 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 244]][z]
Out[7]=	2 4 6 1 + 9 z + 13 z + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 244]}
In[9]:=	{KnotDet[Knot[11, Alternating, 244]], KnotSignature[Knot[11, Alternating, 244]]}
Out[9]=	{147, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 244]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 3 q + 9 q - 14 q + 20 q - 24 q + 24 q - 21 q + 16 q - 10 q + 13 14 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 244]}
In[12]:=	A2Invariant[Knot[11, Alternating, 244]][q]
Out[12]=	10 12 14 16 18 20 22 24 26 30 q - 2 q + 4 q - q + q + 5 q - 4 q + 4 q - 3 q + 2 q - 32 34 36 38 40 42 > 4 q + 3 q - 3 q - q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 244]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 6 -3 3 -6 z 5 z 12 z 3 z z 11 z 3 z z 3 z --- + -- + a - --- - ---- + ----- + ---- - --- + ----- + ---- + --- + ---- + 10 8 12 10 8 6 12 8 6 10 8 a a a a a a a a a a a 6 z > -- 6 a
In[14]:=	Kauffman[Knot[11, Alternating, 244]][a, z]
Out[14]=	2 2 2 2 2 3 3 -6 4 z 5 z 7 z 6 z z 3 z 10 z 11 z 3 z --- + -- - a - --- - --- - --- - --- + --- - ---- - ----- - ----- + ---- - 10 8 15 13 11 9 16 14 10 8 6 a a a a a a a a a a a 3 3 3 3 3 3 4 4 4 4 z 10 z 20 z 25 z 19 z 3 z 4 z 9 z 17 z 22 z > --- + ----- + ----- + ----- + ----- + ---- - ---- + ---- + ----- + ----- + 17 15 13 11 9 7 16 14 12 10 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 15 z 3 z z 14 z 24 z 20 z 17 z 6 z 4 z 16 z > ----- - ---- + --- - ----- - ----- - ----- - ----- - ---- + ---- - ----- - 8 6 17 15 13 11 9 7 16 14 a a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 36 z 31 z 14 z z 9 z 3 z 10 z z 3 z 11 z > ----- - ----- - ----- + -- + ---- + ---- - ----- - -- + ---- + ----- + 12 10 8 6 15 13 11 9 7 14 a a a a a a a a a a 8 8 8 9 9 9 10 10 17 z 12 z 6 z 7 z 12 z 5 z 2 z 2 z > ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 12 10 8 13 11 9 12 10 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 244]], Vassiliev[3][Knot[11, Alternating, 244]]}
Out[15]=	{9, 26}
In[16]:=	Kh[Knot[11, Alternating, 244]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 3 q t + 6 q t + 3 q t + 8 q t + 6 q t + 12 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 8 q t + 12 q t + 12 q t + 12 q t + 12 q t + 9 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 12 q t + 7 q t + 9 q t + 3 q t + 7 q t + q t + 27 10 29 11 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a244

K11a243 K11a245