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

K11n132 K11n134

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X5,19,6,18 X7,20,8,21 X2,10,3,9 X11,17,12,16 X13,6,14,7 X15,9,16,8 X17,1,18,22 X19,15,20,14 X21,12,22,13

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-4 - 4t-3 + 6t-2 - 2t-1 - 1 - 2t + 6t2 - 4t3 + t4

Conway Polynomial:

1 + 2z2 + 2z4 + 4z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{25, 4}

Jones Polynomial:

- q-1 + 3 - 3q + 4q2 - 4q3 + 4q4 - 3q5 + 2q6 - q7

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

{K11n50, K11n132, ...}

A2 (sl(3)) Invariant:

- q-2 + 1 + q2 + q4 + q6 + q8 + q10 - 2q12 + q14 - q16 + q18 - 2q26 + q28

HOMFLY-PT Polynomial:

a-8z2 - a-6 - 5a-6z2 - 5a-6z4 - a-6z6 + a-4 + 8a-4z2 + 11a-4z4 + 6a-4z6 + a-4z8 + a-2 - 2a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

- a-8z2 - 3a-7z + 6a-7z3 - 5a-7z5 + a-7z7 + a-6 - 11a-6z2 + 23a-6z4 - 16a-6z6 + 3a-6z8 - 5a-5z + 10a-5z3 + 2a-5z5 - 8a-5z7 + 2a-5z9 + a-4 - 16a-4z2 + 42a-4z4 - 31a-4z6 + 6a-4z8 - 3a-3z + 7a-3z3 + 3a-3z5 - 8a-3z7 + 2a-3z9 - a-2 - 6a-2z2 + 19a-2z4 - 15a-2z6 + 3a-2z8 - a-1z + 3a-1z3 - 4a-1z5 + a-1z7

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

{2, 3}

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 11133. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 15                 1 j = 13               2 1 j = 11             2 1   j = 9           2 2 1   j = 7         3 3       j = 5       2 2         j = 3     2 4 1         j = 1   1 1             j = -1   2               j = -3 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, 133]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 133]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 19, 6, 18], X[7, 20, 8, 21], > X[2, 10, 3, 9], X[11, 17, 12, 16], X[13, 6, 14, 7], X[15, 9, 16, 8], > X[17, 1, 18, 22], X[19, 15, 20, 14], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 133]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 7, -4, 8, 5, -2, -6, 11, -7, 10, -8, 6, -9, 3, -10, > 4, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 133]]
Out[5]=	DTCode[4, 10, -18, -20, 2, -16, -6, -8, -22, -14, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 133]][t]
Out[6]=	-4 4 6 2 2 3 4 -1 + t - -- + -- - - - 2 t + 6 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 133]][z]
Out[7]=	2 4 6 8 1 + 2 z + 2 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 133]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 133]], KnotSignature[Knot[11, NonAlternating, 133]]}
Out[9]=	{25, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 133]][q]
Out[10]=	1 2 3 4 5 6 7 3 - - - 3 q + 4 q - 4 q + 4 q - 3 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 50], Knot[11, NonAlternating, 132], > Knot[11, NonAlternating, 133]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 133]][q]
Out[12]=	-2 2 4 6 8 10 12 14 16 18 26 28 1 - q + q + q + q + q + q - 2 q + q - q + q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 133]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 6 -6 -4 -2 z 5 z 8 z 2 z 5 z 11 z 4 z z 6 z -a + a + a + -- - ---- + ---- - ---- - ---- + ----- - ---- - -- + ---- - 8 6 4 2 6 4 2 6 4 a a a a a a a a a 6 8 z z > -- + -- 2 4 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 133]][a, z]
Out[14]=	2 2 2 2 3 -6 -4 -2 3 z 5 z 3 z z z 11 z 16 z 6 z 6 z a + a - a - --- - --- - --- - - - -- - ----- - ----- - ---- + ---- + 7 5 3 a 8 6 4 2 7 a a a a a a a a 3 3 3 4 4 4 5 5 5 5 10 z 7 z 3 z 23 z 42 z 19 z 5 z 2 z 3 z 4 z > ----- + ---- + ---- + ----- + ----- + ----- - ---- + ---- + ---- - ---- - 5 3 a 6 4 2 7 5 3 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 9 16 z 31 z 15 z z 8 z 8 z z 3 z 6 z 3 z 2 z > ----- - ----- - ----- + -- - ---- - ---- + -- + ---- + ---- + ---- + ---- + 6 4 2 7 5 3 a 6 4 2 5 a a a a a a a a a a 9 2 z > ---- 3 a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 133]], Vassiliev[3][Knot[11, NonAlternating, 133]]}
Out[15]=	{2, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 133]][q, t]
Out[16]=	3 3 5 1 2 q q 2 q 3 5 7 7 2 4 q + 2 q + ----- + ---- + -- + - + ---- + q t + 2 q t + 3 q t + 3 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 9 4 11 4 13 4 13 5 15 5 > 2 q t + 2 q t + 2 q t + q t + q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n133

K11n132 K11n134