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

K11a1 K11a3

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,12,19,11 X6,14,7,13 X20,16,21,15 X12,18,13,17 X22,20,1,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 14 2 18 6 20 12 22 16

Alexander Polynomial:

- 3t-3 + 15t-2 - 31t-1 + 39 - 31t + 15t2 - 3t3

Conway Polynomial:

1 + 2z2 - 3z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11a116, ...}

Determinant and Signature:

{137, 4}

Jones Polynomial:

1 - 3q + 8q2 - 13q3 + 19q4 - 22q5 + 22q6 - 20q7 + 15q8 - 9q9 + 4q10 - q11

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

{K11a116, ...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + 3q6 - 3q8 + 5q10 - q12 - q14 + 2q16 - 5q18 + 3q20 - 3q22 + q24 + 3q26 - 3q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 3a-8 + 6a-8z2 + 3a-8z4 - 4a-6 - 7a-6z2 - 6a-6z4 - 2a-6z6 + 2a-4 + 2a-4z2 - a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + 2a-12z2 - 5a-12z4 + 4a-12z6 - 3a-11z + 8a-11z3 - 12a-11z5 + 8a-11z7 + a-10 - a-10z2 + 2a-10z4 - 10a-10z6 + 9a-10z8 - 8a-9z + 30a-9z3 - 36a-9z5 + 10a-9z7 + 5a-9z9 + 3a-8 - 14a-8z2 + 35a-8z4 - 45a-8z6 + 20a-8z8 + a-8z10 - 10a-7z + 33a-7z3 - 33a-7z5 + 9a-7z9 + 4a-6 - 18a-6z2 + 38a-6z4 - 42a-6z6 + 16a-6z8 + a-6z10 - 6a-5z + 17a-5z3 - 17a-5z5 + a-5z7 + 4a-5z9 + 2a-4 - 4a-4z2 + 7a-4z4 - 10a-4z6 + 5a-4z8 - a-3z + 5a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{2, 4}

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 r = 8 r = 9 j = 23                       1 j = 21                     3   j = 19                   6 1   j = 17                 9 3     j = 15               11 6       j = 13             11 9         j = 11           11 11           j = 9         8 11             j = 7       5 11               j = 5     3 8                 j = 3   1 6                   j = 1   2                     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, Alternating, 2]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 2]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 8, 15, 7], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[6, 14, 7, 13], X[20, 16, 21, 15], > X[12, 18, 13, 17], X[22, 20, 1, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 2]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -9, 7, -4, 8, -11, 9, -6, 10, > -8, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 2]]
Out[5]=	DTCode[4, 8, 10, 14, 2, 18, 6, 20, 12, 22, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 2]][t]
Out[6]=	3 15 31 2 3 39 - -- + -- - -- - 31 t + 15 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 2]][z]
Out[7]=	2 4 6 1 + 2 z - 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 2], Knot[11, Alternating, 116]}
In[9]:=	{KnotDet[Knot[11, Alternating, 2]], KnotSignature[Knot[11, Alternating, 2]]}
Out[9]=	{137, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 2]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 8 q - 13 q + 19 q - 22 q + 22 q - 20 q + 15 q - 9 q + 4 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 2], Knot[11, Alternating, 116]}
In[12]:=	A2Invariant[Knot[11, Alternating, 2]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 22 1 - q + q + 3 q - 3 q + 5 q - q - q + 2 q - 5 q + 3 q - 3 q + 24 26 28 30 34 > q + 3 q - 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 2]][a, z]
Out[13]=	2 2 2 2 2 4 4 -10 3 4 2 -2 z 6 z 7 z 2 z 2 z 3 z 6 z -a + -- - -- + -- + a - --- + ---- - ---- + ---- + ---- + ---- - ---- - 8 6 4 10 8 6 4 2 8 6 a a a a a a a a a a 4 4 6 6 z z 2 z z > -- + -- - ---- - -- 4 2 6 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 2]][a, z]
Out[14]=	2 2 2 -10 3 4 2 -2 3 z 8 z 10 z 6 z z 2 z z 14 z a + -- + -- + -- - a - --- - --- - ---- - --- - -- + ---- - --- - ----- - 8 6 4 11 9 7 5 3 12 10 8 a a a a a a a a a a a 2 2 2 3 3 3 3 3 3 4 18 z 4 z 3 z z 8 z 30 z 33 z 17 z 5 z 5 z > ----- - ---- + ---- - --- + ---- + ----- + ----- + ----- + ---- - ---- + 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 2 z 35 z 38 z 7 z 3 z z 12 z 36 z 33 z 17 z > ---- + ----- + ----- + ---- - ---- + --- - ----- - ----- - ----- - ----- - 10 8 6 4 2 13 11 9 7 5 a a a a a a a a a a 5 6 6 6 6 6 6 7 7 7 7 z 4 z 10 z 45 z 42 z 10 z z 8 z 10 z z > ---- + ---- - ----- - ----- - ----- - ----- + -- + ---- + ----- + -- + 3 12 10 8 6 4 2 11 9 5 a a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 3 z 9 z 20 z 16 z 5 z 5 z 9 z 4 z z z > ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- + --- + --- 3 10 8 6 4 9 7 5 8 6 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 2]], Vassiliev[3][Knot[11, Alternating, 2]]}
Out[15]=	{2, 4}
In[16]:=	Kh[Knot[11, Alternating, 2]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 6 q + 3 q + ---- + --- + -- + 8 q t + 5 q t + 11 q t + 8 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 11 q t + 11 q t + 11 q t + 11 q t + 9 q t + 11 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 6 q t + 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a2

K11a1 K11a3