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

K11n25 K11n27

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,12,19,11 X13,7,14,6 X20,15,21,16 X22,17,1,18 X10,20,11,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 5t-2 - 9t-1 + 11 - 9t + 5t2 - t3

Conway Polynomial:

1 + 2z2 - z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{920, 10149, ...}

Determinant and Signature:

{41, 0}

Jones Polynomial:

q-2 - 3q-1 + 5 - 6q + 7q2 - 6q3 + 6q4 - 4q5 + 2q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-6 - q-4 + q-2 - q2 + q4 - q6 + 2q8 + q10 + q12 + 2q14 - q16 - q20 - q22

HOMFLY-PT Polynomial:

- 2a-6 - a-6z2 + 4a-4 + 6a-4z2 + 2a-4z4 - 2a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + 1 + 2z2 + z4

Kauffman Polynomial:

- 4a-7z + 8a-7z3 - 5a-7z5 + a-7z7 + 2a-6 - 6a-6z2 + 12a-6z4 - 9a-6z6 + 2a-6z8 - 7a-5z + 19a-5z3 - 13a-5z5 + a-5z9 + 4a-4 - 11a-4z2 + 20a-4z4 - 19a-4z6 + 5a-4z8 - 5a-3z + 16a-3z3 - 16a-3z5 + 2a-3z7 + a-3z9 + 2a-2 - 7a-2z2 + 9a-2z4 - 9a-2z6 + 3a-2z8 - 3a-1z + 8a-1z3 - 8a-1z5 + 3a-1z7 + 1 - z2 + z4 + z6 - az + 3az3 + a2z2

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=0 is the signature of 1126. 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 j = 15                   1 j = 13                 1   j = 11               3 1   j = 9             3 1     j = 7           3 3       j = 5         4 3         j = 3       2 3           j = 1     3 4             j = -1   1 3               j = -3   2                 j = -5 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, 26]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 26]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], > X[14, 9, 15, 10], X[18, 12, 19, 11], X[13, 7, 14, 6], X[20, 15, 21, 16], > X[22, 17, 1, 18], X[10, 20, 11, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 26]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -10, 6, 3, -7, -5, 8, -11, 9, -6, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 26]]
Out[5]=	DTCode[4, 8, -12, 2, 14, 18, -6, 20, 22, 10, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 26]][t]
Out[6]=	-3 5 9 2 3 11 - t + -- - - - 9 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 26]][z]
Out[7]=	2 4 6 1 + 2 z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 20], Knot[10, 149], Knot[11, NonAlternating, 26]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 26]], KnotSignature[Knot[11, NonAlternating, 26]]}
Out[9]=	{41, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 26]][q]
Out[10]=	-2 3 2 3 4 5 6 7 5 + q - - - 6 q + 7 q - 6 q + 6 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 26]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 26]][q]
Out[12]=	-6 -4 -2 2 4 6 8 10 12 14 16 20 22 q - q + q - q + q - q + 2 q + q + q + 2 q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 26]][a, z]
Out[13]=	2 2 2 4 4 6 2 4 2 2 z 6 z 5 z 4 2 z 4 z z 1 - -- + -- - -- + 2 z - -- + ---- - ---- + z + ---- - ---- - -- 6 4 2 6 4 2 4 2 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 26]][a, z]
Out[14]=	2 2 2 2 4 2 4 z 7 z 5 z 3 z 2 6 z 11 z 7 z 1 + -- + -- + -- - --- - --- - --- - --- - a z - z - ---- - ----- - ---- + 6 4 2 7 5 3 a 6 4 2 a a a a a a a a a 3 3 3 3 4 4 4 2 2 8 z 19 z 16 z 8 z 3 4 12 z 20 z 9 z > a z + ---- + ----- + ----- + ---- + 3 a z + z + ----- + ----- + ---- - 7 5 3 a 6 4 2 a a a a a a 5 5 5 5 6 6 6 7 7 7 5 z 13 z 16 z 8 z 6 9 z 19 z 9 z z 2 z 3 z > ---- - ----- - ----- - ---- + z - ---- - ----- - ---- + -- + ---- + ---- + 7 5 3 a 6 4 2 7 3 a a a a a a a a a 8 8 8 9 9 2 z 5 z 3 z z z > ---- + ---- + ---- + -- + -- 6 4 2 5 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 26]], Vassiliev[3][Knot[11, NonAlternating, 26]]}
Out[15]=	{2, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 26]][q, t]
Out[16]=	3 1 2 1 3 3 2 5 2 5 3 - + 3 q + ----- + ---- + --- + 4 q t + 2 q t + 3 q t + 4 q t + 3 q t + q 5 2 3 q t q t q t 7 3 7 4 9 4 9 5 11 5 11 6 13 6 15 7 > 3 q t + 3 q t + 3 q t + q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n26

K11n25 K11n27