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

K11n21 K11n23

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,11,19,12 X13,7,14,6 X20,16,21,15 X10,17,11,18 X22,20,1,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 -12 2 14 18 -6 20 10 22 16

Alexander Polynomial:

t-3 - 5t-2 + 13t-1 - 17 + 13t - 5t2 + t3

Conway Polynomial:

1 + 2z2 + z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{931, K11n11, K11n112, K11n127, ...}

Determinant and Signature:

{55, 2}

Jones Polynomial:

- 1 + 4q - 6q2 + 9q3 - 9q4 + 9q5 - 8q6 + 5q7 - 3q8 + q9

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

{K11n127, ...}

A2 (sl(3)) Invariant:

- 1 + 2q2 - q4 + q6 + 3q8 + 3q12 - q14 - q18 - 3q20 + q22 - q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 4a-6 - 5a-6z2 - 2a-6z4 + 4a-4 + 7a-4z2 + 4a-4z4 + a-4z6 - a-2z2 - a-2z4

Kauffman Polynomial:

2a-10z2 - 3a-10z4 + a-10z6 - 3a-9z + 9a-9z3 - 10a-9z5 + 3a-9z7 + a-8 + a-8z4 - 7a-8z6 + 3a-8z8 - 10a-7z + 25a-7z3 - 24a-7z5 + 5a-7z7 + a-7z9 + 4a-6 - 11a-6z2 + 17a-6z4 - 17a-6z6 + 6a-6z8 - 10a-5z + 24a-5z3 - 18a-5z5 + 4a-5z7 + a-5z9 + 4a-4 - 12a-4z2 + 17a-4z4 - 9a-4z6 + 3a-4z8 - 4a-3z + 9a-3z3 - 4a-3z5 + 2a-3z7 - 3a-2z2 + 4a-2z4 - a-1z + a-1z3

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

{2, 2}

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=2 is the signature of 1122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                   1 j = 17                 2   j = 15               3 1   j = 13             5 2     j = 11           4 3       j = 9         5 5         j = 7       4 4           j = 5     2 5             j = 3   2 4               j = 1   3                 j = -1 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, NonAlternating, 22]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 22]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], > X[14, 9, 15, 10], X[18, 11, 19, 12], X[13, 7, 14, 6], X[20, 16, 21, 15], > X[10, 17, 11, 18], X[22, 20, 1, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 22]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -9, 6, 3, -7, -5, 8, -11, 9, -6, 10, > -8, 11, -10]
In[5]:=	DTCode[Knot[11, NonAlternating, 22]]
Out[5]=	DTCode[4, 8, -12, 2, 14, 18, -6, 20, 10, 22, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 22]][t]
Out[6]=	-3 5 13 2 3 -17 + t - -- + -- + 13 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 22]][z]
Out[7]=	2 4 6 1 + 2 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 31], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 22], > Knot[11, NonAlternating, 112], Knot[11, NonAlternating, 127]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 22]], KnotSignature[Knot[11, NonAlternating, 22]]}
Out[9]=	{55, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 22]][q]
Out[10]=	2 3 4 5 6 7 8 9 -1 + 4 q - 6 q + 9 q - 9 q + 9 q - 8 q + 5 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 22], Knot[11, NonAlternating, 127]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 22]][q]
Out[12]=	2 4 6 8 12 14 18 20 22 24 28 -1 + 2 q - q + q + 3 q + 3 q - q - q - 3 q + q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 22]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 4 4 z 5 z 7 z z 2 z 4 z z z a - -- + -- + -- - ---- + ---- - -- - ---- + ---- - -- + -- 6 4 8 6 4 2 6 4 2 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 22]][a, z]
Out[14]=	2 2 2 2 -8 4 4 3 z 10 z 10 z 4 z z 2 z 11 z 12 z 3 z a + -- + -- - --- - ---- - ---- - --- - - + ---- - ----- - ----- - ---- + 6 4 9 7 5 3 a 10 6 4 2 a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 9 z 25 z 24 z 9 z z 3 z z 17 z 17 z 4 z > ---- + ----- + ----- + ---- + -- - ---- + -- + ----- + ----- + ---- - 9 7 5 3 a 10 8 6 4 2 a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 10 z 24 z 18 z 4 z z 7 z 17 z 9 z 3 z 5 z > ----- - ----- - ----- - ---- + --- - ---- - ----- - ---- + ---- + ---- + 9 7 5 3 10 8 6 4 9 7 a a a a a a a a a a 7 7 8 8 8 9 9 4 z 2 z 3 z 6 z 3 z z z > ---- + ---- + ---- + ---- + ---- + -- + -- 5 3 8 6 4 7 5 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 22]], Vassiliev[3][Knot[11, NonAlternating, 22]]}
Out[15]=	{2, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 22]][q, t]
Out[16]=	3 1 3 5 5 2 7 2 7 3 9 3 3 q + 2 q + --- + 4 q t + 2 q t + 5 q t + 4 q t + 4 q t + 5 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 15 7 > 5 q t + 4 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 17 7 19 8 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n22

K11n21 K11n23