K11a57

K11a56

K11a58

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a57 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X16,6,17,5 X2837 X20,9,21,10 X22,11,1,12 X18,13,19,14 X6,16,7,15 X12,17,13,18 X14,19,15,20 X10,21,11,22
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -10, 8, -3, 9, -7, 10, -5, 11, -6
Dowker-Thistlethwaite code	4 8 16 2 20 22 18 6 12 14 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a57/ThurstonBennequinNumber Hyperbolic Volume 14.7354 A-Polynomial See Data:K11a57/A-polynomial

[edit Notes for K11a57's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ [2,4] }[/math] Rasmussen s-Invariant 2

[edit Notes for K11a57's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-12 t^2+20 t-23+20 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{t^2-t+1\right\} }[/math]
Determinant and Signature	{ 99, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+2 q^3-5 q^2+10 q-12+16 q^{-1} -16 q^{-2} +14 q^{-3} -12 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-15 a^2 z^4-z^4 a^{-2} +10 z^4+6 a^4 z^2-19 a^2 z^2-4 z^2 a^{-2} +18 z^2+3 a^4-10 a^2-4 a^{-2} +12 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^{10}+z^{10}+4 a^3 z^9+6 a z^9+2 z^9 a^{-1} +8 a^4 z^8+10 a^2 z^8+2 z^8 a^{-2} +4 z^8+9 a^5 z^7+4 a^3 z^7-9 a z^7-3 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-12 a^4 z^6-31 a^2 z^6-8 z^6 a^{-2} -21 z^6+3 a^7 z^5-16 a^5 z^5-29 a^3 z^5-14 a z^5-9 z^5 a^{-1} -5 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+6 a^4 z^4+30 a^2 z^4+11 z^4 a^{-2} +28 z^4-2 a^7 z^3+16 a^5 z^3+35 a^3 z^3+29 a z^3+20 z^3 a^{-1} +8 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-2 a^4 z^2-22 a^2 z^2-8 z^2 a^{-2} -24 z^2-6 a^5 z-16 a^3 z-17 a z-11 z a^{-1} -4 z a^{-3} +3 a^4+10 a^2+4 a^{-2} +12 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}-q^{18}+3 q^{16}-q^{14}-q^{12}-6 q^8+q^6-3 q^4+4 q^2+5+2 q^{-2} +4 q^{-4} -2 q^{-6} - q^{-8} - q^{-10} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+6 q^{106}-5 q^{104}+9 q^{100}-19 q^{98}+29 q^{96}-38 q^{94}+35 q^{92}-22 q^{90}-5 q^{88}+49 q^{86}-87 q^{84}+115 q^{82}-119 q^{80}+82 q^{78}-16 q^{76}-81 q^{74}+178 q^{72}-232 q^{70}+223 q^{68}-133 q^{66}-4 q^{64}+156 q^{62}-252 q^{60}+268 q^{58}-177 q^{56}+20 q^{54}+136 q^{52}-227 q^{50}+204 q^{48}-65 q^{46}-111 q^{44}+242 q^{42}-274 q^{40}+170 q^{38}+19 q^{36}-243 q^{34}+375 q^{32}-397 q^{30}+265 q^{28}-43 q^{26}-210 q^{24}+377 q^{22}-414 q^{20}+315 q^{18}-126 q^{16}-99 q^{14}+262 q^{12}-302 q^{10}+229 q^8-54 q^6-119 q^4+242 q^2-219+99 q^{-2} +86 q^{-4} -233 q^{-6} +304 q^{-8} -242 q^{-10} +84 q^{-12} +106 q^{-14} -243 q^{-16} +304 q^{-18} -249 q^{-20} +121 q^{-22} +17 q^{-24} -131 q^{-26} +174 q^{-28} -164 q^{-30} +104 q^{-32} -36 q^{-34} -21 q^{-36} +51 q^{-38} -59 q^{-40} +45 q^{-42} -26 q^{-44} +8 q^{-46} +2 q^{-48} -9 q^{-50} +7 q^{-52} -6 q^{-54} +4 q^{-56} - q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

K11a57 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a57"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-12 t^2+20 t-23+20 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-3 z^6-2 z^4+z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \left\{t^2-t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 99, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^4+2 q^3-5 q^2+10 q-12+16 q^{-1} -16 q^{-2} +14 q^{-3} -12 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-15 a^2 z^4-z^4 a^{-2} +10 z^4+6 a^4 z^2-19 a^2 z^2-4 z^2 a^{-2} +18 z^2+3 a^4-10 a^2-4 a^{-2} +12 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ a^2 z^{10}+z^{10}+4 a^3 z^9+6 a z^9+2 z^9 a^{-1} +8 a^4 z^8+10 a^2 z^8+2 z^8 a^{-2} +4 z^8+9 a^5 z^7+4 a^3 z^7-9 a z^7-3 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-12 a^4 z^6-31 a^2 z^6-8 z^6 a^{-2} -21 z^6+3 a^7 z^5-16 a^5 z^5-29 a^3 z^5-14 a z^5-9 z^5 a^{-1} -5 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+6 a^4 z^4+30 a^2 z^4+11 z^4 a^{-2} +28 z^4-2 a^7 z^3+16 a^5 z^3+35 a^3 z^3+29 a z^3+20 z^3 a^{-1} +8 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-2 a^4 z^2-22 a^2 z^2-8 z^2 a^{-2} -24 z^2-6 a^5 z-16 a^3 z-17 a z-11 z a^{-1} -4 z a^{-3} +3 a^4+10 a^2+4 a^{-2} +12 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a108, K11a139, K11a231,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a231,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a57"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ -t^4+5 t^3-12 t^2+20 t-23+20 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+2 q^3-5 q^2+10 q-12+16 q^{-1} -16 q^{-2} +14 q^{-3} -12 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11a108, K11a139, K11a231,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a231,}

Vassiliev invariants

V2 and V3:

(1, 3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 4 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ \frac{58}{3} }[/math] [math]\displaystyle{ 96 }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{56}{3} }[/math] [math]\displaystyle{ \frac{232}{3} }[/math] [math]\displaystyle{ \frac{19231}{30} }[/math] [math]\displaystyle{ \frac{1258}{15} }[/math] [math]\displaystyle{ \frac{13622}{45} }[/math] [math]\displaystyle{ \frac{929}{18} }[/math] [math]\displaystyle{ \frac{511}{30} }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of K11a57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     1   1 5                   4 1   -3 3                 6 1     5 1               6 4       -2 -1             10 6         4 -3           7 7           0 -5         7 9             -2 -7       5 7               2 -9     2 7                 -5 -11   1 5                   4 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a56 K11a58