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

K11n119 K11n121

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-4 + 4t-3 - 7t-2 + 8t-1 - 7 + 8t - 7t2 + 4t3 - t4

Conway Polynomial:

1 - 3z4 - 4z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{47, 2}

Jones Polynomial:

q-3 - 3q-2 + 5q-1 - 6 + 8q - 8q2 + 7q3 - 5q4 + 3q5 - q6

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

{...}

A2 (sl(3)) Invariant:

q-8 - q-6 + q-4 + 1 + q2 - 2q4 + 2q6 - 2q8 + 2q10 + q16 - q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 3a-4 + 7a-4z2 + 5a-4z4 + a-4z6 - 3a-2 - 10a-2z2 - 12a-2z4 - 6a-2z6 - a-2z8 + 2 + 4z2 + 4z4 + z6

Kauffman Polynomial:

- a-7z + a-7z3 + a-6 - 4a-6z2 + 3a-6z4 - 2a-5z + 3a-5z3 - a-5z5 + a-5z7 + 3a-4 - 16a-4z2 + 25a-4z4 - 13a-4z6 + 3a-4z8 - 2a-3z + 3a-3z3 + 5a-3z5 - 6a-3z7 + 2a-3z9 + 3a-2 - 20a-2z2 + 40a-2z4 - 29a-2z6 + 7a-2z8 - 2a-1z + 7a-1z3 - 4a-1z5 - 4a-1z7 + 2a-1z9 + 2 - 7z2 + 15z4 - 15z6 + 4z8 - az + 6az3 - 10az5 + 3az7 + a2z2 - 3a2z4 + a2z6

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

{0, 1}

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 11120. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   1 j = 11                 2   j = 9               3 1   j = 7             4 2     j = 5           4 3       j = 3         4 4         j = 1       3 5           j = -1     2 3             j = -3   1 3               j = -5   2                 j = -7 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, 120]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 120]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 15, 6, 14], X[7, 20, 8, 21], > X[2, 10, 3, 9], X[11, 17, 12, 16], X[13, 18, 14, 19], X[15, 9, 16, 8], > X[17, 1, 18, 22], X[19, 6, 20, 7], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 120]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 10, -4, 8, 5, -2, -6, 11, -7, 3, -8, 6, -9, 7, -10, > 4, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 120]]
Out[5]=	DTCode[4, 10, -14, -20, 2, -16, -18, -8, -22, -6, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 120]][t]
Out[6]=	-4 4 7 8 2 3 4 -7 - t + -- - -- + - + 8 t - 7 t + 4 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 120]][z]
Out[7]=	4 6 8 1 - 3 z - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 120]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 120]], KnotSignature[Knot[11, NonAlternating, 120]]}
Out[9]=	{47, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 120]][q]
Out[10]=	-3 3 5 2 3 4 5 6 -6 + q - -- + - + 8 q - 8 q + 7 q - 5 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 120]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 120]][q]
Out[12]=	-8 -6 -4 2 4 6 8 10 16 18 20 22 1 + q - q + q + q - 2 q + 2 q - 2 q + 2 q + q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 120]][a, z]
Out[13]=	2 2 2 4 4 6 -6 3 3 2 z 7 z 10 z 4 5 z 12 z 6 z 2 - a + -- - -- + 4 z - -- + ---- - ----- + 4 z + ---- - ----- + z + -- - 4 2 6 4 2 4 2 4 a a a a a a a a 6 8 6 z z > ---- - -- 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 120]][a, z]
Out[14]=	2 2 2 -6 3 3 z 2 z 2 z 2 z 2 4 z 16 z 20 z 2 + a + -- + -- - -- - --- - --- - --- - a z - 7 z - ---- - ----- - ----- + 4 2 7 5 3 a 6 4 2 a a a a a a a a 3 3 3 3 4 4 4 2 2 z 3 z 3 z 7 z 3 4 3 z 25 z 40 z > a z + -- + ---- + ---- + ---- + 6 a z + 15 z + ---- + ----- + ----- - 7 5 3 a 6 4 2 a a a a a a 5 5 5 6 6 7 2 4 z 5 z 4 z 5 6 13 z 29 z 2 6 z > 3 a z - -- + ---- - ---- - 10 a z - 15 z - ----- - ----- + a z + -- - 5 3 a 4 2 5 a a a a a 7 7 8 8 9 9 6 z 4 z 7 8 3 z 7 z 2 z 2 z > ---- - ---- + 3 a z + 4 z + ---- + ---- + ---- + ---- 3 a 4 2 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 120]], Vassiliev[3][Knot[11, NonAlternating, 120]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 120]][q, t]
Out[16]=	3 1 2 1 3 2 3 3 q 3 5 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 4 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n120

K11n119 K11n121