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

K11a240 K11a242

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 18 20 22 16 2 10 12 8 6

Alexander Polynomial:

4t-3 - 12t-2 + 20t-1 - 23 + 20t - 12t2 + 4t3

Conway Polynomial:

1 + 8z2 + 12z4 + 4z6

Other knots with the same Alexander/Conway Polynomial:

{K11a186, ...}

Determinant and Signature:

{95, 6}

Jones Polynomial:

q3 - 2q4 + 6q5 - 9q6 + 13q7 - 15q8 + 15q9 - 14q10 + 10q11 - 6q12 + 3q13 - q14

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

{K11a186, ...}

A2 (sl(3)) Invariant:

q10 - q12 + 3q14 + q18 + 3q20 - 2q22 + 3q24 - 2q26 - q28 - 3q32 + 2q34 - q36 + q40 - q42

HOMFLY-PT Polynomial:

- 2a-12z2 - a-12z4 - 2a-10 - a-10z2 + 2a-10z4 + a-10z6 + a-8 + 6a-8z2 + 7a-8z4 + 2a-8z6 + 2a-6 + 5a-6z2 + 4a-6z4 + a-6z6

Kauffman Polynomial:

- 2a-17z3 + a-17z5 + 2a-16z2 - 6a-16z4 + 3a-16z6 - 3a-15z + 9a-15z3 - 11a-15z5 + 5a-15z7 - a-14z2 + 7a-14z4 - 9a-14z6 + 5a-14z8 - 4a-13z + 12a-13z3 - 8a-13z5 + 3a-13z9 + a-12z2 + 6a-12z4 - 10a-12z6 + 5a-12z8 + a-12z10 - 6a-11z + 9a-11z3 - 2a-11z5 - 6a-11z7 + 5a-11z9 + 2a-10 - 4a-10z2 + 2a-10z4 - 6a-10z6 + 3a-10z8 + a-10z10 - 4a-9z + 10a-9z3 - 11a-9z5 + a-9z7 + 2a-9z9 + a-8 - 3a-8z2 + 5a-8z4 - 7a-8z6 + 3a-8z8 + a-7z + 2a-7z3 - 5a-7z5 + 2a-7z7 - 2a-6 + 5a-6z2 - 4a-6z4 + a-6z6

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

{8, 22}

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=6 is the signature of 11241. 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 r = 10 r = 11 j = 29                       1 j = 27                     2   j = 25                   4 1   j = 23                 6 2     j = 21               8 4       j = 19             7 6         j = 17           8 8           j = 15         5 7             j = 13       4 8               j = 11     2 5                 j = 9     4                   j = 7 1 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, Alternating, 241]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 241]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[20, 8, 21, 7], > X[22, 10, 1, 9], X[16, 12, 17, 11], X[2, 14, 3, 13], X[10, 16, 11, 15], > X[12, 18, 13, 17], X[8, 20, 9, 19], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 241]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -9, 7, -2, 8, -6, 9, -3, 10, > -4, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 241]]
Out[5]=	DTCode[4, 14, 18, 20, 22, 16, 2, 10, 12, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 241]][t]
Out[6]=	4 12 20 2 3 -23 + -- - -- + -- + 20 t - 12 t + 4 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 241]][z]
Out[7]=	2 4 6 1 + 8 z + 12 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 186], Knot[11, Alternating, 241]}
In[9]:=	{KnotDet[Knot[11, Alternating, 241]], KnotSignature[Knot[11, Alternating, 241]]}
Out[9]=	{95, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 241]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 2 q + 6 q - 9 q + 13 q - 15 q + 15 q - 14 q + 10 q - 6 q + 13 14 > 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 186], Knot[11, Alternating, 241]}
In[12]:=	A2Invariant[Knot[11, Alternating, 241]][q]
Out[12]=	10 12 14 18 20 22 24 26 28 32 34 q - q + 3 q + q + 3 q - 2 q + 3 q - 2 q - q - 3 q + 2 q - 36 40 42 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 241]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 -2 -8 2 2 z z 6 z 5 z z 2 z 7 z 4 z z --- + a + -- - ---- - --- + ---- + ---- - --- + ---- + ---- + ---- + --- + 10 6 12 10 8 6 12 10 8 6 10 a a a a a a a a a a a 6 6 2 z z > ---- + -- 8 6 a a
In[14]:=	Kauffman[Knot[11, Alternating, 241]][a, z]
Out[14]=	2 2 2 2 2 2 -8 2 3 z 4 z 6 z 4 z z 2 z z z 4 z 3 z --- + a - -- - --- - --- - --- - --- + -- + ---- - --- + --- - ---- - ---- + 10 6 15 13 11 9 7 16 14 12 10 8 a a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 5 z 2 z 9 z 12 z 9 z 10 z 2 z 6 z 7 z 6 z > ---- - ---- + ---- + ----- + ---- + ----- + ---- - ---- + ---- + ---- + 6 17 15 13 11 9 7 16 14 12 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 2 z 5 z 4 z z 11 z 8 z 2 z 11 z 5 z 3 z > ---- + ---- - ---- + --- - ----- - ---- - ---- - ----- - ---- + ---- - 10 8 6 17 15 13 11 9 7 16 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 8 8 9 z 10 z 6 z 7 z z 5 z 6 z z 2 z 5 z 5 z > ---- - ----- - ---- - ---- + -- + ---- - ---- + -- + ---- + ---- + ---- + 14 12 10 8 6 15 11 9 7 14 12 a a a a a a a a a a a 8 8 9 9 9 10 10 3 z 3 z 3 z 5 z 2 z z z > ---- + ---- + ---- + ---- + ---- + --- + --- 10 8 13 11 9 12 10 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 241]], Vassiliev[3][Knot[11, Alternating, 241]]}
Out[15]=	{8, 22}
In[16]:=	Kh[Knot[11, Alternating, 241]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 2 q t + 4 q t + 2 q t + 5 q t + 4 q t + 8 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 5 q t + 7 q t + 8 q t + 8 q t + 7 q t + 6 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 8 q t + 4 q t + 6 q t + 2 q t + 4 q t + q t + 27 10 29 11 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a241

K11a240 K11a242