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

K11n1 K11n3

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,15,8,14 X2,9,3,10 X16,12,17,11 X20,14,21,13 X15,7,16,6 X22,18,1,17 X12,20,13,19 X18,22,19,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-3 + 8t-2 - 12t-1 + 13 - 12t + 8t2 - 2t3

Conway Polynomial:

1 + 2z2 - 4z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1014, K11a161, ...}

Determinant and Signature:

{57, 4}

Jones Polynomial:

1 - 2q + 5q2 - 7q3 + 9q4 - 10q5 + 9q6 - 7q7 + 5q8 - 2q9

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

{...}

A2 (sl(3)) Invariant:

1 + q4 + 2q6 - q8 + 2q10 - 2q12 - 2q18 + 2q20 - q22 + 2q24 + q26 - q28 - q32

HOMFLY-PT Polynomial:

- a-10 + 2a-8 + 3a-8z2 + a-8z4 - a-6 - 2a-6z2 - 3a-6z4 - a-6z6 - a-4 - 2a-4z2 - 3a-4z4 - a-4z6 + 2a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

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

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

{2, 5}

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=4 is the signature of 112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                   2 j = 17                 3   j = 15               4 2   j = 13             5 3     j = 11           5 4       j = 9         4 5         j = 7       3 5           j = 5     2 4             j = 3   1 4               j = 1   1                 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, 2]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 2]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 15, 8, 14], > X[2, 9, 3, 10], X[16, 12, 17, 11], X[20, 14, 21, 13], X[15, 7, 16, 6], > X[22, 18, 1, 17], X[12, 20, 13, 19], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 2]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -10, 7, 4, -8, -6, 9, -11, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 2]]
Out[5]=	DTCode[4, 8, 10, -14, 2, 16, 20, -6, 22, 12, 18]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 2]][t]
Out[6]=	2 8 12 2 3 13 - -- + -- - -- - 12 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 2]][z]
Out[7]=	2 4 6 1 + 2 z - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 14], Knot[11, Alternating, 161], Knot[11, NonAlternating, 2]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 2]], KnotSignature[Knot[11, NonAlternating, 2]]}
Out[9]=	{57, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 2]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 2 q + 5 q - 7 q + 9 q - 10 q + 9 q - 7 q + 5 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 2]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 2]][q]
Out[12]=	4 6 8 10 12 18 20 22 24 26 28 1 + q + 2 q - q + 2 q - 2 q - 2 q + 2 q - q + 2 q + q - q - 32 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 2]][a, z]
Out[13]=	2 2 2 2 4 4 4 -10 2 -6 -4 2 3 z 2 z 2 z 3 z z 3 z 3 z -a + -- - a - a + -- + ---- - ---- - ---- + ---- + -- - ---- - ---- + 8 2 8 6 4 2 8 6 4 a a a a a a a a a 4 6 6 z z z > -- - -- - -- 2 6 4 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 2]][a, z]
Out[14]=	2 2 2 2 2 -10 2 -6 -4 2 2 z 2 z 5 z 10 z 5 z 5 z 5 z a + -- + a - a - -- - --- - --- - ---- - ----- - ---- + ---- + ---- + 8 2 11 9 10 8 6 4 2 a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 3 z 6 z 3 z 4 z 4 z 4 z 12 z 8 z 4 z 4 z > ---- + ---- + ---- + ---- + ---- + ---- + ----- + ---- - ---- - ---- - 11 9 7 5 3 10 8 6 4 2 a a a a a a a a a a 5 5 5 5 6 6 6 6 6 7 7 4 z 7 z 9 z 6 z z 6 z 11 z 3 z z 3 z 3 z > ---- - ---- - ---- - ---- + --- - ---- - ----- - ---- + -- + ---- + ---- + 9 7 5 3 10 8 6 4 2 9 7 a a a a a a a a a a a 7 7 8 8 8 9 9 2 z 2 z 3 z 5 z 2 z z z > ---- + ---- + ---- + ---- + ---- + -- + -- 5 3 8 6 4 7 5 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 2]], Vassiliev[3][Knot[11, NonAlternating, 2]]}
Out[15]=	{2, 5}
In[16]:=	Kh[Knot[11, NonAlternating, 2]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 4 q t + 3 q t + 5 q t + 4 q t + 5 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 5 q t + 4 q t + 5 q t + 3 q t + 4 q t + 2 q t + 17 6 19 7 > 3 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n2

K11n1 K11n3