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

K11a26 K11a28

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X12,6,13,5 X14,7,15,8 X2,9,3,10 X18,11,19,12 X22,14,1,13 X20,15,21,16 X10,17,11,18 X16,19,17,20 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 14 2 18 22 20 10 16 6

Alexander Polynomial:

2t-3 - 13t-2 + 34t-1 - 45 + 34t - 13t2 + 2t3

Conway Polynomial:

1 - z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{143, -2}

Jones Polynomial:

q-9 - 4q-8 + 9q-7 - 15q-6 + 20q-5 - 23q-4 + 23q-3 - 20q-2 + 15q-1 - 8 + 4q - q2

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-26 - 2q-24 + 4q-22 - 2q-20 + 3q-16 - 5q-14 + 2q-12 - 3q-10 + q-8 + 3q-6 - 3q-4 + 6q-2 - 1 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + 3a2 + 4a2z2 + 2a2z4 + a2z6 - 3a4 - 3a4z2 + a4z6 + a6 - a6z2 - 2a6z4 + a8z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 3z2 - 6z4 + 4z6 - az + 4az3 - 9az5 + 7az7 - 3a2 + 9a2z2 - 6a2z4 - 6a2z6 + 8a2z8 - 2a3z + 12a3z3 - 16a3z5 + 2a3z7 + 6a3z9 - 3a4 + 6a4z2 + 11a4z4 - 29a4z6 + 14a4z8 + 2a4z10 - 2a5z + 10a5z3 - 7a5z5 - 14a5z7 + 12a5z9 - a6 - 3a6z2 + 22a6z4 - 35a6z6 + 13a6z8 + 2a6z10 - 2a7z + 9a7z3 - 10a7z5 - 5a7z7 + 6a7z9 - 2a8z2 + 9a8z4 - 15a8z6 + 7a8z8 - a9z + 6a9z3 - 9a9z5 + 4a9z7 + a10z2 - 2a10z4 + a10z6

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 1127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     3   j = 1                   5 1   j = -1                 10 3     j = -3               11 6       j = -5             12 9         j = -7           11 11           j = -9         9 12             j = -11       6 11               j = -13     3 9                 j = -15   1 6                   j = -17   3                     j = -19 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, 27]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 27]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[22, 14, 1, 13], X[20, 15, 21, 16], > X[10, 17, 11, 18], X[16, 19, 17, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 27]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -3, 7, -4, 8, -10, 9, -6, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 27]]
Out[5]=	DTCode[4, 8, 12, 14, 2, 18, 22, 20, 10, 16, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 27]][t]
Out[6]=	2 13 34 2 3 -45 + -- - -- + -- + 34 t - 13 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 27]][z]
Out[7]=	4 6 1 - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 27]}
In[9]:=	{KnotDet[Knot[11, Alternating, 27]], KnotSignature[Knot[11, Alternating, 27]]}
Out[9]=	{143, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 27]][q]
Out[10]=	-9 4 9 15 20 23 23 20 15 2 -8 + q - -- + -- - -- + -- - -- + -- - -- + -- + 4 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 27]}
In[12]:=	A2Invariant[Knot[11, Alternating, 27]][q]
Out[12]=	-28 -26 2 4 2 3 5 2 3 -8 3 3 -1 + q - q - --- + --- - --- + --- - --- + --- - --- + q + -- - -- + 24 22 20 16 14 12 10 6 4 q q q q q q q q q 6 2 4 6 > -- - q + 2 q - q 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 27]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 8 2 4 2 4 3 a - 3 a + a - z + 4 a z - 3 a z - a z + a z - z + 2 a z - 6 4 2 6 4 6 > 2 a z + a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 27]][a, z]
Out[14]=	2 4 6 3 5 7 9 2 2 2 -3 a - 3 a - a - a z - 2 a z - 2 a z - 2 a z - a z + 3 z + 9 a z + 3 4 2 6 2 8 2 10 2 z 3 3 3 5 3 > 6 a z - 3 a z - 2 a z + a z - -- + 4 a z + 12 a z + 10 a z + a 7 3 9 3 4 2 4 4 4 6 4 8 4 > 9 a z + 6 a z - 6 z - 6 a z + 11 a z + 22 a z + 9 a z - 5 10 4 z 5 3 5 5 5 7 5 9 5 6 > 2 a z + -- - 9 a z - 16 a z - 7 a z - 10 a z - 9 a z + 4 z - a 2 6 4 6 6 6 8 6 10 6 7 3 7 > 6 a z - 29 a z - 35 a z - 15 a z + a z + 7 a z + 2 a z - 5 7 7 7 9 7 2 8 4 8 6 8 8 8 > 14 a z - 5 a z + 4 a z + 8 a z + 14 a z + 13 a z + 7 a z + 3 9 5 9 7 9 4 10 6 10 > 6 a z + 12 a z + 6 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 27]], Vassiliev[3][Knot[11, Alternating, 27]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, Alternating, 27]][q, t]
Out[16]=	6 10 1 3 1 6 3 9 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 11 9 12 11 11 12 9 11 3 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a27

K11a26 K11a28