K11a28

K11a27

K11a29

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a28 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X12,6,13,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,14,21,13 X22,16,1,15 X10,17,11,18 X6,19,7,20 X14,22,15,21
Gauss code	1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -3, 7, -11, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 12 16 2 18 20 22 10 6 14

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,2\}} 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a28/ThurstonBennequinNumber Hyperbolic Volume 15.8799 A-Polynomial See Data:K11a28/A-polynomial

[edit Notes for K11a28's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,4]} Rasmussen s-Invariant 0

[edit Notes for K11a28's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8+2 z^6-z^4-2 z^2+1}
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+13q^{2}-17q+20-19q^{-1}+16q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +9 z^4+3 a^4 z^2-10 a^2 z^2-2 z^2 a^{-2} +7 z^2+2 a^4-4 a^2+3}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+11 a z^9+6 z^9 a^{-1} +5 a^4 z^8+6 a^2 z^8+8 z^8 a^{-2} +9 z^8+3 a^5 z^7-11 a^3 z^7-28 a z^7-7 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-28 a^2 z^6-13 z^6 a^{-2} +4 z^6 a^{-4} -31 z^6-8 a^5 z^5+9 a^3 z^5+31 a z^5+2 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+11 a^4 z^4+38 a^2 z^4+7 z^4 a^{-2} -6 z^4 a^{-4} +37 z^4+5 a^5 z^3-6 a^3 z^3-15 a z^3+3 z^3 a^{-3} -z^3 a^{-5} +2 a^6 z^2-7 a^4 z^2-22 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -16 z^2-a^5 z+a^3 z+3 a z+z a^{-1} +2 a^4+4 a^2+3}
The A2 invariant	$q^{18}+2q^{12}-3q^{10}+2q^{8}-2q^{6}-2q^{4}+3q^{2}-3+5q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+11 q^{86}-12 q^{84}+5 q^{82}+11 q^{80}-33 q^{78}+61 q^{76}-81 q^{74}+79 q^{72}-47 q^{70}-21 q^{68}+120 q^{66}-215 q^{64}+274 q^{62}-251 q^{60}+121 q^{58}+88 q^{56}-314 q^{54}+476 q^{52}-483 q^{50}+327 q^{48}-38 q^{46}-283 q^{44}+499 q^{42}-523 q^{40}+337 q^{38}-25 q^{36}-273 q^{34}+422 q^{32}-359 q^{30}+115 q^{28}+202 q^{26}-448 q^{24}+502 q^{22}-341 q^{20}-3 q^{18}+378 q^{16}-653 q^{14}+716 q^{12}-528 q^{10}+163 q^8+264 q^6-606 q^4+735 q^2-612+288 q^{-2} +107 q^{-4} -413 q^{-6} +520 q^{-8} -381 q^{-10} +100 q^{-12} +206 q^{-14} -387 q^{-16} +362 q^{-18} -157 q^{-20} -144 q^{-22} +397 q^{-24} -489 q^{-26} +401 q^{-28} -161 q^{-30} -121 q^{-32} +341 q^{-34} -440 q^{-36} +397 q^{-38} -250 q^{-40} +59 q^{-42} +109 q^{-44} -215 q^{-46} +243 q^{-48} -203 q^{-50} +132 q^{-52} -44 q^{-54} -30 q^{-56} +72 q^{-58} -88 q^{-60} +73 q^{-62} -46 q^{-64} +21 q^{-66} +2 q^{-68} -13 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }

Further Quantum Invariants

K11a28 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["K11a28"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8+2 z^6-z^4-2 z^2+1}

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+13q^{2}-17q+20-19q^{-1}+16q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +9 z^4+3 a^4 z^2-10 a^2 z^2-2 z^2 a^{-2} +7 z^2+2 a^4-4 a^2+3}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+11 a z^9+6 z^9 a^{-1} +5 a^4 z^8+6 a^2 z^8+8 z^8 a^{-2} +9 z^8+3 a^5 z^7-11 a^3 z^7-28 a z^7-7 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-28 a^2 z^6-13 z^6 a^{-2} +4 z^6 a^{-4} -31 z^6-8 a^5 z^5+9 a^3 z^5+31 a z^5+2 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+11 a^4 z^4+38 a^2 z^4+7 z^4 a^{-2} -6 z^4 a^{-4} +37 z^4+5 a^5 z^3-6 a^3 z^3-15 a z^3+3 z^3 a^{-3} -z^3 a^{-5} +2 a^6 z^2-7 a^4 z^2-22 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -16 z^2-a^5 z+a^3 z+3 a z+z a^{-1} +2 a^4+4 a^2+3}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_123,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a87, K11a96,}

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["K11a28"];

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]=

{ $t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^5+4 q^4-8 q^3+13 q^2-17 q+20-19 q^{-1} +16 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} } }

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]=

{10_123,}

In[6]:=

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

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

Out[6]=

{K11a87, K11a96,}

Vassiliev invariants

V2 and V3:

(-2, 2)

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16} $32$ ${\frac {116}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{76}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -128} $-{\frac {704}{3}}$ $-{\frac {320}{3}}$ $-16$ $-{\frac {256}{3}}$ $128$ $-{\frac {928}{3}}$ $-{\frac {608}{3}}$ ${\frac {1289}{15}}$ ${\frac {388}{5}}$ $-{\frac {5884}{45}}$ ${\frac {391}{9}}$ $-{\frac {871}{15}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or $j-2r=s-1$, where $s=$0 is the signature of K11a28. 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 11                       1 -1 9                     3   3 7                   5 1   -4 5                 8 3     5 3               9 5       -4 1             11 8         3 -1           9 10           1 -3         7 10             -3 -5       5 9               4 -7     2 7                 -5 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a27 K11a29