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

K11n143 K11n145

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X18,6,19,5 X7,15,8,14 X16,10,17,9 X2,12,3,11 X22,13,1,14 X15,20,16,21 X10,18,11,17 X6,20,7,19 X21,9,22,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 15t-1 + 19 - 15t + 7t2 - t3

Conway Polynomial:

1 + 4z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n10, K11n103, ...}

Determinant and Signature:

{65, 4}

Jones Polynomial:

2q2 - 4q3 + 8q4 - 10q5 + 11q6 - 11q7 + 9q8 - 6q9 + 3q10 - q11

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

{K11n10, ...}

A2 (sl(3)) Invariant:

2q6 - q8 + 3q10 + q12 - q14 + 2q16 - 3q18 + q20 - q22 + 2q26 - 2q28 + q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 2a-8 + 4a-8z2 + 2a-8z4 - 3a-6 - 4a-6z2 - 3a-6z4 - a-6z6 + 3a-4 + 5a-4z2 + 2a-4z4

Kauffman Polynomial:

- 2a-13z3 + a-13z5 + 2a-12z2 - 6a-12z4 + 3a-12z6 - 4a-11z + 10a-11z3 - 12a-11z5 + 5a-11z7 + a-10 - 3a-10z2 + 5a-10z4 - 7a-10z6 + 4a-10z8 - 6a-9z + 21a-9z3 - 21a-9z5 + 7a-9z7 + a-9z9 + 2a-8 - 9a-8z2 + 16a-8z4 - 13a-8z6 + 6a-8z8 - 4a-7z + 8a-7z3 - 7a-7z5 + 3a-7z7 + a-7z9 + 3a-6 - 10a-6z2 + 8a-6z4 - 3a-6z6 + 2a-6z8 - 2a-5z - a-5z3 + a-5z5 + a-5z7 + 3a-4 - 6a-4z2 + 3a-4z4

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

{4, 9}

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 11144. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj 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                 2   j = 19               4 1   j = 17             5 2     j = 15           6 4       j = 13         5 5         j = 11       5 6           j = 9     3 5             j = 7   1 5               j = 5 1 3                 j = 3 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, 144]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 144]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[7, 15, 8, 14], > X[16, 10, 17, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[15, 20, 16, 21], > X[10, 18, 11, 17], X[6, 20, 7, 19], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 144]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, -4, 11, 5, -9, 6, -2, 7, 4, -8, -5, 9, -3, 10, > 8, -11, -7]
In[5]:=	DTCode[Knot[11, NonAlternating, 144]]
Out[5]=	DTCode[4, 12, 18, -14, 16, 2, 22, -20, 10, 6, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 144]][t]
Out[6]=	-3 7 15 2 3 19 - t + -- - -- - 15 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 144]][z]
Out[7]=	2 4 6 1 + 4 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 10], Knot[11, NonAlternating, 103], > Knot[11, NonAlternating, 144]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 144]], KnotSignature[Knot[11, NonAlternating, 144]]}
Out[9]=	{65, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 144]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 2 q - 4 q + 8 q - 10 q + 11 q - 11 q + 9 q - 6 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 10], Knot[11, NonAlternating, 144]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 144]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 26 28 2 q - q + 3 q + q - q + 2 q - 3 q + q - q + 2 q - 2 q + 30 34 > q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 144]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 2 3 3 z 4 z 4 z 5 z 2 z 3 z 2 z z -a + -- - -- + -- - --- + ---- - ---- + ---- + ---- - ---- + ---- - -- 8 6 4 10 8 6 4 8 6 4 6 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 144]][a, z]
Out[14]=	2 2 2 2 -10 2 3 3 4 z 6 z 4 z 2 z 2 z 3 z 9 z 10 z a + -- + -- + -- - --- - --- - --- - --- + ---- - ---- - ---- - ----- - 8 6 4 11 9 7 5 12 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 6 z 2 z 10 z 21 z 8 z z 6 z 5 z 16 z 8 z > ---- - ---- + ----- + ----- + ---- - -- - ---- + ---- + ----- + ---- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 3 z z 12 z 21 z 7 z z 3 z 7 z 13 z 3 z > ---- + --- - ----- - ----- - ---- + -- + ---- - ---- - ----- - ---- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 5 z 7 z 3 z z 4 z 6 z 2 z z z > ---- + ---- + ---- + -- + ---- + ---- + ---- + -- + -- 11 9 7 5 10 8 6 9 7 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 144]], Vassiliev[3][Knot[11, NonAlternating, 144]]}
Out[15]=	{4, 9}
In[16]:=	Kh[Knot[11, NonAlternating, 144]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 11 4 2 q + q + 3 q t + q t + 5 q t + 3 q t + 5 q t + 5 q t + 6 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 5 q t + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 19 7 19 8 21 8 23 9 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n144

K11n143 K11n145