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

K11a228 K11a230

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X20,6,21,5 X18,8,19,7 X14,10,15,9 X2,12,3,11 X10,14,11,13 X22,15,1,16 X8,18,9,17 X6,20,7,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 20 18 14 2 10 22 8 6 16

Alexander Polynomial:

- 4t-2 + 18t-1 - 27 + 18t - 4t2

Conway Polynomial:

1 + 2z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{K11a226, ...}

Determinant and Signature:

{71, 2}

Jones Polynomial:

q-1 - 2 + 5q - 8q2 + 10q3 - 11q4 + 11q5 - 9q6 + 7q7 - 4q8 + 2q9 - q10

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

{...}

A2 (sl(3)) Invariant:

q-4 + 3q2 - 2q4 + q8 - 2q10 + q12 - q14 + q16 + q18 - q20 + 3q22 - q26 + q28 - q30 - q32

HOMFLY-PT Polynomial:

- a-10 + a-8 + 2a-8z2 + a-6 + a-6z2 - a-6z4 - a-4 - 2a-4z2 - 2a-4z4 - a-2z4 + 1 + z2

Kauffman Polynomial:

- 2a-11z + 7a-11z3 - 5a-11z5 + a-11z7 + a-10 - 5a-10z2 + 12a-10z4 - 9a-10z6 + 2a-10z8 - a-9z3 + 6a-9z5 - 7a-9z7 + 2a-9z9 + a-8 - 2a-8z2 + 2a-8z4 - 3a-8z6 - a-8z8 + a-8z10 + 2a-7z - 12a-7z3 + 14a-7z5 - 12a-7z7 + 4a-7z9 - a-6 + 8a-6z2 - 12a-6z4 + 2a-6z6 + a-6z10 - 2a-5z + 5a-5z3 - 5a-5z5 + 2a-5z9 - a-4 + 7a-4z2 - 5a-4z4 - a-4z6 + 3a-4z8 - 2a-3z + 7a-3z3 - 6a-3z5 + 4a-3z7 - 2a-2z4 + 3a-2z6 - 2a-1z3 + 2a-1z5 + 1 - 2z2 + z4

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

{2, 7}

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 11229. 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 = 21                       1 j = 19                     1   j = 17                   3 1   j = 15                 4 1     j = 13               5 3       j = 11             6 4         j = 9           5 5           j = 7         5 6             j = 5       3 5               j = 3     2 5                 j = 1   1 4                   j = -1   1                     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, Alternating, 229]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 229]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 6, 21, 5], X[18, 8, 19, 7], > X[14, 10, 15, 9], X[2, 12, 3, 11], X[10, 14, 11, 13], X[22, 15, 1, 16], > X[8, 18, 9, 17], X[6, 20, 7, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 229]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -7, 6, -2, 7, -5, 8, -11, 9, -4, 10, > -3, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 229]]
Out[5]=	DTCode[4, 12, 20, 18, 14, 2, 10, 22, 8, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 229]][t]
Out[6]=	4 18 2 -27 - -- + -- + 18 t - 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 229]][z]
Out[7]=	2 4 1 + 2 z - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 226], Knot[11, Alternating, 229]}
In[9]:=	{KnotDet[Knot[11, Alternating, 229]], KnotSignature[Knot[11, Alternating, 229]]}
Out[9]=	{71, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 229]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 -2 + - + 5 q - 8 q + 10 q - 11 q + 11 q - 9 q + 7 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 229]}
In[12]:=	A2Invariant[Knot[11, Alternating, 229]][q]
Out[12]=	-4 2 4 8 10 12 14 16 18 20 22 26 q + 3 q - 2 q + q - 2 q + q - q + q + q - q + 3 q - q + 28 30 32 > q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 229]][a, z]
Out[13]=	2 2 2 4 4 4 -10 -8 -6 -4 2 2 z z 2 z z 2 z z 1 - a + a + a - a + z + ---- + -- - ---- - -- - ---- - -- 8 6 4 6 4 2 a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 229]][a, z]
Out[14]=	2 2 -10 -8 -6 -4 2 z 2 z 2 z 2 z 2 5 z 2 z 1 + a + a - a - a - --- + --- - --- - --- - 2 z - ---- - ---- + 11 7 5 3 10 8 a a a a a a 2 2 3 3 3 3 3 3 4 4 8 z 7 z 7 z z 12 z 5 z 7 z 2 z 4 12 z 2 z > ---- + ---- + ---- - -- - ----- + ---- + ---- - ---- + z + ----- + ---- - 6 4 11 9 7 5 3 a 10 8 a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 12 z 5 z 2 z 5 z 6 z 14 z 5 z 6 z 2 z 9 z > ----- - ---- - ---- - ---- + ---- + ----- - ---- - ---- + ---- - ---- - 6 4 2 11 9 7 5 3 a 10 a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 8 3 z 2 z z 3 z z 7 z 12 z 4 z 2 z z 3 z > ---- + ---- - -- + ---- + --- - ---- - ----- + ---- + ---- - -- + ---- + 8 6 4 2 11 9 7 3 10 8 4 a a a a a a a a a a a 9 9 9 10 10 2 z 4 z 2 z z z > ---- + ---- + ---- + --- + --- 9 7 5 8 6 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 229]], Vassiliev[3][Knot[11, Alternating, 229]]}
Out[15]=	{2, 7}
In[16]:=	Kh[Knot[11, Alternating, 229]][q, t]
Out[16]=	3 1 1 q 3 5 5 2 7 2 7 3 4 q + 2 q + ----- + --- + - + 5 q t + 3 q t + 5 q t + 5 q t + 6 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 5 q t + 5 q t + 6 q t + 4 q t + 5 q t + 3 q t + 4 q t + 15 7 17 7 17 8 19 8 21 9 > q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a229

K11a228 K11a230