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

K11n39 K11n41

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-3 - 8t-2 + 18t-1 - 23 + 18t - 8t2 + 2t3

Conway Polynomial:

1 + 4z2 + 4z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1057, K11n46, ...}

Determinant and Signature:

{79, 2}

Jones Polynomial:

- 2 + 6q - 9q2 + 13q3 - 13q4 + 13q5 - 11q6 + 7q7 - 4q8 + q9

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

{K11n46, ...}

A2 (sl(3)) Invariant:

- 2 + 2q2 - 2q4 + q6 + 4q8 + 5q12 - q14 + q16 - q18 - 4q20 + q22 - 2q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 6a-6 - 8a-6z2 - 3a-6z4 + 8a-4 + 15a-4z2 + 9a-4z4 + 2a-4z6 - 2a-2 - 4a-2z2 - 2a-2z4

Kauffman Polynomial:

a-10z2 - 2a-10z4 + a-10z6 - a-9z + 8a-9z3 - 11a-9z5 + 4a-9z7 + a-8 + 3a-8z4 - 11a-8z6 + 5a-8z8 - 9a-7z + 28a-7z3 - 32a-7z5 + 7a-7z7 + 2a-7z9 + 6a-6 - 15a-6z2 + 22a-6z4 - 27a-6z6 + 11a-6z8 - 13a-5z + 32a-5z3 - 30a-5z5 + 8a-5z7 + 2a-5z9 + 8a-4 - 20a-4z2 + 23a-4z4 - 14a-4z6 + 6a-4z8 - 7a-3z + 15a-3z3 - 9a-3z5 + 5a-3z7 + 2a-2 - 6a-2z2 + 6a-2z4 + a-2z6 - 2a-1z + 3a-1z3

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

{4, 6}

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 1140. 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                 3   j = 15               4 1   j = 13             7 3     j = 11           6 4       j = 9         7 7         j = 7       6 6           j = 5     3 7             j = 3   3 6               j = 1   4                 j = -1 2

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, 40]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 40]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], > X[18, 10, 19, 9], X[16, 11, 17, 12], X[13, 7, 14, 6], X[20, 15, 21, 16], > X[22, 18, 1, 17], X[14, 19, 15, 20], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 40]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -11, 6, 3, -7, -10, 8, -6, 9, -5, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 40]]
Out[5]=	DTCode[4, 8, -12, 2, 18, 16, -6, 20, 22, 14, 10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 40]][t]
Out[6]=	2 8 18 2 3 -23 + -- - -- + -- + 18 t - 8 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 40]][z]
Out[7]=	2 4 6 1 + 4 z + 4 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 57], Knot[11, NonAlternating, 40], Knot[11, NonAlternating, 46]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 40]], KnotSignature[Knot[11, NonAlternating, 40]]}
Out[9]=	{79, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 40]][q]
Out[10]=	2 3 4 5 6 7 8 9 -2 + 6 q - 9 q + 13 q - 13 q + 13 q - 11 q + 7 q - 4 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 40], Knot[11, NonAlternating, 46]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 40]][q]
Out[12]=	2 4 6 8 12 14 16 18 20 22 24 -2 + 2 q - 2 q + q + 4 q + 5 q - q + q - q - 4 q + q - 2 q + 28 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 40]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 6 8 2 z 8 z 15 z 4 z 3 z 9 z 2 z 2 z a - -- + -- - -- + -- - ---- + ----- - ---- - ---- + ---- - ---- + ---- 6 4 2 8 6 4 2 6 4 2 4 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 40]][a, z]
Out[14]=	2 2 2 2 -8 6 8 2 z 9 z 13 z 7 z 2 z z 15 z 20 z 6 z a + -- + -- + -- - -- - --- - ---- - --- - --- + --- - ----- - ----- - ---- + 6 4 2 9 7 5 3 a 10 6 4 2 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 8 z 28 z 32 z 15 z 3 z 2 z 3 z 22 z 23 z 6 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 6 7 11 z 32 z 30 z 9 z z 11 z 27 z 14 z z 4 z > ----- - ----- - ----- - ---- + --- - ----- - ----- - ----- + -- + ---- + 9 7 5 3 10 8 6 4 2 9 a a a a a a a a a a 7 7 7 8 8 8 9 9 7 z 8 z 5 z 5 z 11 z 6 z 2 z 2 z > ---- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 7 5 3 8 6 4 7 5 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 40]], Vassiliev[3][Knot[11, NonAlternating, 40]]}
Out[15]=	{4, 6}
In[16]:=	Kh[Knot[11, NonAlternating, 40]][q, t]
Out[16]=	3 2 3 5 5 2 7 2 7 3 9 3 4 q + 3 q + --- + 6 q t + 3 q t + 7 q t + 6 q t + 6 q t + 7 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 15 7 > 7 q t + 6 q t + 4 q t + 7 q t + 3 q t + 4 q t + q t + 17 7 19 8 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n40

K11n39 K11n41